First of all, I must apologise for the fact that I haven't been updating my blog as much as I'd intended to over the past few weeks. As often happens, real life has gotten in the way, including (for example) my graduation. I can now officially carry the post-nominal letters BScAdv(Hons), although anyone concerned for my ever-inflating ego will no doubt be glad to hear that I have very little intention of doing so.
In terms of the blog, I'm currently working on a post on Fourier analysis and the following post will likely be on the physics of mechanical flight, by popular request. At the current rate I'm going that might not be for a while though!
In physics news, since my last news post some rumours have arisen that the BICEP2 results may not be as sound as initially claimed. Basically, Adam Falkowski has claimed that the BICEP2 collaboration have miscalculated the galactic foreground radiation by misinterpreting an image on a Planck collaboration slide, and their primordial polarisations can mostly be accounted for due to this error (a claim that the BICEP2 team strongly denies). A sceptical take on the rumour is provided by Sesh Nadathur, who argues that the issue has been blown far out of proportion. It will be interesting to see how things unfold in the coming months!
Thursday, May 29, 2014
Monday, April 21, 2014
Why Heisenberg uncertainty is not that weird
Whenever quantum mechanics (QM) is brought up in a popular context, in a scientific or pseudo-scientific way, the 'weirdness' of it is almost always mentioned, and the Heisenberg uncertainty principle$^1$ is almost always the go-to example of the weirdness (although in pseudo-scientific contexts it is almost always misrepresented).
So what is Heisenberg uncertainty? Simply put, it is a restriction on the accuracy of simultaneous measurements of 'observables' (measurable quantities). The prototypical example is position and momentum; Heisenberg uncertainty states that the position and momentum of a particle$^2$ cannot be known simultaneously to arbitrary precision. The mathematical statement of the position-momentum uncertainty principle is
\begin{equation}
\Delta x\Delta p\leq\frac{\hbar}{2}
\end{equation}
where $\hbar$ is the reduced Planck constant and $\Delta x$, $\Delta p$ represent (in some sense)$^3$ the uncertainty in $x$ and $p$ respectively. Effectively, the better you know position, the less well you know momentum, and vice versa. This is not a specifically experimental limitation, but a fundamental theoretical one. This sort of restriction, on first viewing, indeed seems very strange and certainly counter-intuitive. I will attempt to convince you that not only is it not necessarily strange, but expected.
What is important to remember about QM is that wave mechanics is a central theme. Particles are represented by wave-functions, which are complex solutions to the Schrödinger equation,$^4$ and this wave-nature contributes to a good deal of the quantum weirdness we are familiar with (an example is shown in a recent blog post of mine which relies on superposition and destructive interference of photon waves). With this in mind, let's take a look at some wave-functions.
For illustrative purposes, we will work in one spatial dimension and free space (zero potential everywhere) and only consider time-independent wave-functions. The simplest example of such a wave-function is the plane wave, which takes the form
\begin{equation}
\psi(x)=Ae^{ikx}\equiv Ae^{ipx/\hbar}.
\end{equation}
Here $A$ is the complex-valued amplitude. The amplitude in this case is not important, because any wave-function must be normalisable (as the probability distribution function, which must integrate to 1 over all space to preserve conservation of probability, is given by $|\psi(x)|^2$, known as the Born rule) and so $A$ will need to be scaled anyway. For those who are unfamiliar with complex exponential form, the waviness is more explicit in the less compact form $\exp{(ikx)}\equiv\cos{(kx)}+i\sin{(kx)}$. The $k$ is the wave-number (or wave-vector in higher dimensions), and this term appears naturally in most of the mathematics I'm presenting in this post. For this reason I will include the version of the equations with $k$ alongside the version with the more physically immediate momentum $p$ (the conversion is simply $p=\hbar k$).
Because the Schrödinger equation is linear, sums of solutions are themselves solutions (this property is known as the superposition principle). That means we can have wave-functions of the form
\begin{equation}
\psi(x)=\sum_{m=0}^{n}A_me^{ip_mx/\hbar}
\end{equation}
for any arbitrary $n$ (finite or infinite). Here the scaling of the $A$ values is still not important because of normalisation, but the relative magnitude of them is, as this determines the relative probability weightings according to the Born rule. However, because we have wavelength $\lambda_m=2\pi/k_m\equiv2\pi\hbar/p_m$, we can see that the above formulation does not capture the full number of possible modes; only integer multiples of the $m=0$ mode are captured. In free space all modes are permissible and so we can take the continuum limit (let the sum turn into an integral):
\begin{equation}
\psi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{\psi}(k) e^{ikx}\mathrm{d}k\equiv\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\tilde{\psi}(p) e^{ipx/\hbar}\mathrm{d}p.
\end{equation}
Because we have moved away from integer indexing, the discrete set of amplitudes $A_m$ is replaced by the continuous function that is suggestively symbolled $\tilde{\psi}$. The function ranges over $p$ because we are integrating over all possible modes/wavelengths/momenta—in a sense $p$ takes over the role of index in the integral from $m$ in the summation. The factor of $1/\sqrt{2\pi}$ is a matter of convention and the factor of $1/\sqrt{\hbar}$ comes from the change of $k$ to $p$.
There is more to $\tilde{\psi}$ than meets the eye. Not only is it the amplitude function for the integral, but it's actually the wave-function itself, except not in physical space like $\psi$ but in momentum space.$^5$ In the context of QM this is known as the momentum space representation of the wave-function, but more broadly the mathematical construct is known as the Fourier transform,$^6$ and Fourier transforms occur very frequently in all manner of physical theories involving waves, be they QM, acoustics, optics, crystallography, signal analysis and so on.
So how do we determine the form of $\tilde{\psi}$? As it turns out, perhaps unsurprisingly, the Fourier transform is invertible, and so we find that
\begin{equation}
\tilde{\psi}(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(x) e^{-ipx/\hbar}\mathrm{d}x.
\end{equation}
All well and good, but how do we make sense of it? Well, let's consider a limiting case. We can select out a single mode by using a Dirac delta such that $\tilde{\psi}(p)=\delta(p-p_0)$. This Dirac delta is zero everywhere except at $p=p_0$, where it is undefined but the area under the Dirac delta is always normalised to $1$. Inserting the delta into equation (4) yields$^7$
\begin{equation}
\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\delta(p-p_0)e^{ipx/\hbar}\mathrm{d}p=\frac{e^{ip_0x/\hbar}}{\sqrt{2\pi\hbar}},
\end{equation}
which is, outside a numerical factor, effectively a complex exponential in $x$, or equivalently, a flat (complex-valued) wave across all space (applying the Born rule to a plane wave yields a probability distribution of $|e^{iz}|^2=\text{const.}$). So when the momentum is maximally well-defined (put into a single mode) the position is maximally poorly-defined (the wave-function is spread evenly across all space). By the invertibility of the Fourier transform, we can expect the vice versa case to hold (a maximally defined position, i.e., Dirac delta in $x$, will result in a maximally poorly-defined momentum, i.e., an even wave across all momentum space). This relationships lies at the heart of the Heisenberg uncertainty principle.
It is easy to see why (physically speaking) a Dirac delta in momentum space produces a plane wave in position space, but perhaps not so for the other way around. One way to think of it is by considering the example of a Gaussian wave-packet in position space, which takes the Gaussian distribution as its probability distribution, making it somewhat localised (it's non-zero across all space, but asymptotes to zero approaching $\pm\infty$). It is not hard to show that the Fourier transform of the Gaussian wave-packet is also a Gaussian wave-packet. As we decrease the width of the position space wave-packet, we expect to require higher-frequency momentum modes as these will have shorter wavelengths and thus allow for more cancellation closer to the peak, therefore the momentum space wave-packet must correspondingly increase in width. Taking the limiting case, as the position space wave-packet becomes increasingly peaked the momentum space wave-packet must become increasingly spread-out, leading to the Dirac delta-plane wave correspondence we expected. This argument can also be used in reverse to achieve the more intuitive result of a Dirac delta in momentum space leading to a plane wave in position space.
As hinted to at that start of the post, position and momentum are not the only observables which obey Heisenberg uncertainty. The next most common pairing$^8$ is energy and time,$^9$ although uncertainty relationships can be generated more generally by taking the derivative of the (classical) action (and quantising). For example, momentum is the derivative of the action with respect to position, energy is the derivative of the action with respect to time, and so on.
Hopefully I have demonstrated that Heisenberg uncertainty is not so strange as might have first appeared. Rather than an arbitrary restriction on how accurately we can know certain measurable quantities, it is in fact a basic and unavoidable feature of any linear wave theory, of which quantum mechanics is only one example, albeit of a more fundamental and therefore perhaps more intuitively challenging sort than most.
$2$. I hesitate to use the word 'particle', as it is important when talking about these quantum concepts to be very clear about what one means. A better term might have been 'quantum' or 'wave-particle', as the wave-nature of QM is central to the discussion of Heisenberg uncertainty. However, despite the slightly misleading connotations, 'particle' is by far the most commonly used term and so I will use it also.
$3$. We could, for example, take the standard deviation in $x$ and $p$, $\sigma_x$ and $\sigma_p$, as these will be represented by continuous distributions.
$4$. In this post I will use the Schrödinger representation of QM as this makes explicit the wave-nature of the wave-function. However, there are many, many representations of QM, each with their own advantages (and disadvantages) when it comes to analysing real systems, but importantly they are all exactly equivalent and so this wave-nature I have been emphasising is intrinsic to all of them. In that sense I could just as well have chosen any representation and this blog post would otherwise have been identical, although perhaps not as easy to understand.
$5$. If you don't know what momentum space is, the most important thing to understand is that there are mathematical spaces other than the space(time) we are familiar with. Let's consider the flat 3-dimensional "position" space (3-space) of ordinary life, with $x$-, $y$- and $z$-directions. Suppose we have an object at the coordinates $x=1$, $y=0$ and $z=0$. This can be represented by a vector in the 3-space going to the point $(1,0,0)$, thus describing the position. Now suppose the 3-space we are looking at is part of a 4-space that includes time, except we are going to set the time to some instant and freeze it there.
Let's say at that instant the object at $(1,0,0)$ is travelling with a momentum of $0$ in the $x$-direction, $1$ in the $y$-direction and $0$ in the $z$-direction (in arbitrary units). We could then construct a "momentum" 3-space (or 4-space including time) with directions $p_x$, $p_y$ and $p_z$ and at that instant of time the vector corresponding to the object would be at the coordinates $(0,1,0)$. So for any $n$-dimensional position space it's easy to see there is a corresponding $n$-dimensional momentum space. In fact, we can define a $2n$-dimensional space known as the phase space by combining the position and momentum spaces, and that space will describe all possible states of a physical system.
$6$. I want to stress that strictly speaking the momentum representation of the wave-function is not the Fourier transform of the position representation. The Fourier transform dual to position is the wave-vector (or wave-number in 1-D) and not the momentum, although given one is a scalar multiple of the other I feel like we can be a little loose in our communication in this one respect.
$7$. In evaluating equation (6) we have made use of the so-called sifting property of the Dirac delta, where $\int_{-\infty}^{\infty}\delta(x-a)f(x)\mathrm{d}x=f(a)$. This property is analogous to the Kronecker delta for sums but used instead of integrals and is arguably its most useful feature.
$8$. These pairs of variables are typically referred to in physics as 'conjugate variables', although in this context we can also call them Fourier transform duals. It is important to remember that we would be less inclined to refer to them as such if we were working in, for example, the Heisenberg representation of QM where the Heisenberg uncertainty principle arises more directly out of the non-commutativity of Hermitian operator matrices. This is only because the Fourier transforms are implicit in that representation; they are still there in some sense due to the equivalence of representations of QM as discussed in Note 4, but are not nearly as obvious.
$9$. The energy-time Heisenberg uncertainty principle is given mathematically as $\Delta E\Delta t=\hbar/2$. This is analogous to position-momentum uncertainty in the sense that the Fourier dual of position is wave-number $k=p/\hbar$ and not momentum directly; the Fourier dual of time is technically angular frequency $\omega=E/\hbar$ and not energy directly. As in the position-momentum case, however, the difference is only a scalar factor of $\hbar$ and so we can speak reasonably loosely with some impunity.
So what is Heisenberg uncertainty? Simply put, it is a restriction on the accuracy of simultaneous measurements of 'observables' (measurable quantities). The prototypical example is position and momentum; Heisenberg uncertainty states that the position and momentum of a particle$^2$ cannot be known simultaneously to arbitrary precision. The mathematical statement of the position-momentum uncertainty principle is
\begin{equation}
\Delta x\Delta p\leq\frac{\hbar}{2}
\end{equation}
where $\hbar$ is the reduced Planck constant and $\Delta x$, $\Delta p$ represent (in some sense)$^3$ the uncertainty in $x$ and $p$ respectively. Effectively, the better you know position, the less well you know momentum, and vice versa. This is not a specifically experimental limitation, but a fundamental theoretical one. This sort of restriction, on first viewing, indeed seems very strange and certainly counter-intuitive. I will attempt to convince you that not only is it not necessarily strange, but expected.
What is important to remember about QM is that wave mechanics is a central theme. Particles are represented by wave-functions, which are complex solutions to the Schrödinger equation,$^4$ and this wave-nature contributes to a good deal of the quantum weirdness we are familiar with (an example is shown in a recent blog post of mine which relies on superposition and destructive interference of photon waves). With this in mind, let's take a look at some wave-functions.
For illustrative purposes, we will work in one spatial dimension and free space (zero potential everywhere) and only consider time-independent wave-functions. The simplest example of such a wave-function is the plane wave, which takes the form
\begin{equation}
\psi(x)=Ae^{ikx}\equiv Ae^{ipx/\hbar}.
\end{equation}
Here $A$ is the complex-valued amplitude. The amplitude in this case is not important, because any wave-function must be normalisable (as the probability distribution function, which must integrate to 1 over all space to preserve conservation of probability, is given by $|\psi(x)|^2$, known as the Born rule) and so $A$ will need to be scaled anyway. For those who are unfamiliar with complex exponential form, the waviness is more explicit in the less compact form $\exp{(ikx)}\equiv\cos{(kx)}+i\sin{(kx)}$. The $k$ is the wave-number (or wave-vector in higher dimensions), and this term appears naturally in most of the mathematics I'm presenting in this post. For this reason I will include the version of the equations with $k$ alongside the version with the more physically immediate momentum $p$ (the conversion is simply $p=\hbar k$).
Because the Schrödinger equation is linear, sums of solutions are themselves solutions (this property is known as the superposition principle). That means we can have wave-functions of the form
\begin{equation}
\psi(x)=\sum_{m=0}^{n}A_me^{ip_mx/\hbar}
\end{equation}
for any arbitrary $n$ (finite or infinite). Here the scaling of the $A$ values is still not important because of normalisation, but the relative magnitude of them is, as this determines the relative probability weightings according to the Born rule. However, because we have wavelength $\lambda_m=2\pi/k_m\equiv2\pi\hbar/p_m$, we can see that the above formulation does not capture the full number of possible modes; only integer multiples of the $m=0$ mode are captured. In free space all modes are permissible and so we can take the continuum limit (let the sum turn into an integral):
\begin{equation}
\psi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{\psi}(k) e^{ikx}\mathrm{d}k\equiv\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\tilde{\psi}(p) e^{ipx/\hbar}\mathrm{d}p.
\end{equation}
Because we have moved away from integer indexing, the discrete set of amplitudes $A_m$ is replaced by the continuous function that is suggestively symbolled $\tilde{\psi}$. The function ranges over $p$ because we are integrating over all possible modes/wavelengths/momenta—in a sense $p$ takes over the role of index in the integral from $m$ in the summation. The factor of $1/\sqrt{2\pi}$ is a matter of convention and the factor of $1/\sqrt{\hbar}$ comes from the change of $k$ to $p$.
There is more to $\tilde{\psi}$ than meets the eye. Not only is it the amplitude function for the integral, but it's actually the wave-function itself, except not in physical space like $\psi$ but in momentum space.$^5$ In the context of QM this is known as the momentum space representation of the wave-function, but more broadly the mathematical construct is known as the Fourier transform,$^6$ and Fourier transforms occur very frequently in all manner of physical theories involving waves, be they QM, acoustics, optics, crystallography, signal analysis and so on.
So how do we determine the form of $\tilde{\psi}$? As it turns out, perhaps unsurprisingly, the Fourier transform is invertible, and so we find that
\begin{equation}
\tilde{\psi}(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(x) e^{-ipx/\hbar}\mathrm{d}x.
\end{equation}
All well and good, but how do we make sense of it? Well, let's consider a limiting case. We can select out a single mode by using a Dirac delta such that $\tilde{\psi}(p)=\delta(p-p_0)$. This Dirac delta is zero everywhere except at $p=p_0$, where it is undefined but the area under the Dirac delta is always normalised to $1$. Inserting the delta into equation (4) yields$^7$
\begin{equation}
\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\delta(p-p_0)e^{ipx/\hbar}\mathrm{d}p=\frac{e^{ip_0x/\hbar}}{\sqrt{2\pi\hbar}},
\end{equation}
which is, outside a numerical factor, effectively a complex exponential in $x$, or equivalently, a flat (complex-valued) wave across all space (applying the Born rule to a plane wave yields a probability distribution of $|e^{iz}|^2=\text{const.}$). So when the momentum is maximally well-defined (put into a single mode) the position is maximally poorly-defined (the wave-function is spread evenly across all space). By the invertibility of the Fourier transform, we can expect the vice versa case to hold (a maximally defined position, i.e., Dirac delta in $x$, will result in a maximally poorly-defined momentum, i.e., an even wave across all momentum space). This relationships lies at the heart of the Heisenberg uncertainty principle.
 |
| In this figure, the blue curve is the probability distribution for a Gaussian wave-packet and the red curve is the Fourier transform of the blue curve. All sub-figures are to the same (arbitrary) scale. (i) The blue curve is given by $|\exp{(-x^2+ix)}|^2$. (ii) The blue curve is given by $|\exp{(-(2x)^2+i(2x))}|^2$. (iii) The blue curve is given by $|\exp{(-(4x)^2+i(4x))}|^2$. It is clear to see that as the width of the blue curve is decreased, the width of its Fourier transform correspondingly increases. |
As hinted to at that start of the post, position and momentum are not the only observables which obey Heisenberg uncertainty. The next most common pairing$^8$ is energy and time,$^9$ although uncertainty relationships can be generated more generally by taking the derivative of the (classical) action (and quantising). For example, momentum is the derivative of the action with respect to position, energy is the derivative of the action with respect to time, and so on.
Hopefully I have demonstrated that Heisenberg uncertainty is not so strange as might have first appeared. Rather than an arbitrary restriction on how accurately we can know certain measurable quantities, it is in fact a basic and unavoidable feature of any linear wave theory, of which quantum mechanics is only one example, albeit of a more fundamental and therefore perhaps more intuitively challenging sort than most.
Notes
$1$. The Heisenberg uncertainty principle is so ubiquitous in quantum physics that it is frequently referred to simply as 'the uncertainty principle'. Out of habit, however, I tend to use the less common 'Heisenberg uncertainty' to differentiate it from other, admittedly much less common uncertainty relations. As far as I know, any of these uses are considered acceptable.$2$. I hesitate to use the word 'particle', as it is important when talking about these quantum concepts to be very clear about what one means. A better term might have been 'quantum' or 'wave-particle', as the wave-nature of QM is central to the discussion of Heisenberg uncertainty. However, despite the slightly misleading connotations, 'particle' is by far the most commonly used term and so I will use it also.
$3$. We could, for example, take the standard deviation in $x$ and $p$, $\sigma_x$ and $\sigma_p$, as these will be represented by continuous distributions.
$4$. In this post I will use the Schrödinger representation of QM as this makes explicit the wave-nature of the wave-function. However, there are many, many representations of QM, each with their own advantages (and disadvantages) when it comes to analysing real systems, but importantly they are all exactly equivalent and so this wave-nature I have been emphasising is intrinsic to all of them. In that sense I could just as well have chosen any representation and this blog post would otherwise have been identical, although perhaps not as easy to understand.
$5$. If you don't know what momentum space is, the most important thing to understand is that there are mathematical spaces other than the space(time) we are familiar with. Let's consider the flat 3-dimensional "position" space (3-space) of ordinary life, with $x$-, $y$- and $z$-directions. Suppose we have an object at the coordinates $x=1$, $y=0$ and $z=0$. This can be represented by a vector in the 3-space going to the point $(1,0,0)$, thus describing the position. Now suppose the 3-space we are looking at is part of a 4-space that includes time, except we are going to set the time to some instant and freeze it there.
Let's say at that instant the object at $(1,0,0)$ is travelling with a momentum of $0$ in the $x$-direction, $1$ in the $y$-direction and $0$ in the $z$-direction (in arbitrary units). We could then construct a "momentum" 3-space (or 4-space including time) with directions $p_x$, $p_y$ and $p_z$ and at that instant of time the vector corresponding to the object would be at the coordinates $(0,1,0)$. So for any $n$-dimensional position space it's easy to see there is a corresponding $n$-dimensional momentum space. In fact, we can define a $2n$-dimensional space known as the phase space by combining the position and momentum spaces, and that space will describe all possible states of a physical system.
$6$. I want to stress that strictly speaking the momentum representation of the wave-function is not the Fourier transform of the position representation. The Fourier transform dual to position is the wave-vector (or wave-number in 1-D) and not the momentum, although given one is a scalar multiple of the other I feel like we can be a little loose in our communication in this one respect.
$7$. In evaluating equation (6) we have made use of the so-called sifting property of the Dirac delta, where $\int_{-\infty}^{\infty}\delta(x-a)f(x)\mathrm{d}x=f(a)$. This property is analogous to the Kronecker delta for sums but used instead of integrals and is arguably its most useful feature.
$8$. These pairs of variables are typically referred to in physics as 'conjugate variables', although in this context we can also call them Fourier transform duals. It is important to remember that we would be less inclined to refer to them as such if we were working in, for example, the Heisenberg representation of QM where the Heisenberg uncertainty principle arises more directly out of the non-commutativity of Hermitian operator matrices. This is only because the Fourier transforms are implicit in that representation; they are still there in some sense due to the equivalence of representations of QM as discussed in Note 4, but are not nearly as obvious.
$9$. The energy-time Heisenberg uncertainty principle is given mathematically as $\Delta E\Delta t=\hbar/2$. This is analogous to position-momentum uncertainty in the sense that the Fourier dual of position is wave-number $k=p/\hbar$ and not momentum directly; the Fourier dual of time is technically angular frequency $\omega=E/\hbar$ and not energy directly. As in the position-momentum case, however, the difference is only a scalar factor of $\hbar$ and so we can speak reasonably loosely with some impunity.
Thursday, April 10, 2014
News (2014/04/10)
Two items of news to report this week (well, one and a half at least). The half-piece of news is that I'm working on a new blog post which will hopefully be posted some time next week (although it might take until the week after).
The real news is that the LHC has confirmed the existence of Z(4430), a so-called "exotic hadron". Hadrons are composite particles made of quarks (and held together by gluons). According to the quark model, hadrons can only form in one of two ways: a quark-antiquark pairing (known as a meson) and in a quark triplet (know as a baryon).
The most common hadrons in the universe are protons and neutrons which form atomic nuclei. Protons consist of 2 up-type quarks and 1 down-type quark ("uud") while neutrons consist of 1 up quark and 2 down quarks ("udd"). The names and details of quark types, which are known as "flavours", is not something I will go into here, as it is an interesting enough topic to deserve its own post (although a proper explanation for the layperson would need a little more than that I think).
The quark model is a simple one though and does not describe all of the dynamics permitted by quantum chromodynamics ("QCD", the part of the Standard Model that describes strong interactions). This leaves open the door for exotic hadrons which are not mesons or baryons. Z(4430) is one such exotic hadron.
It was first 'discovered' in 2007 (although 5-sigma confirmation didn't come until 2008) and has now been observed at the LHCb experiment at 13.9-sigma accuracy. This means the chances of the observation being a statistical fluke are $1$ in $1.579\times10^{43}$ (a very, very, very large number indeed). It is believed to be a tetraquark made up of 1 charm quark, 1 charm antiquark, 1 down quark and 1 up antiquark ("ccdu").
While perhaps not as exciting as, for example, the BICEP2 result recently, this confirmation is still a very interesting result and will hopefully spur on further developments in the search for exotic hadrons.
The real news is that the LHC has confirmed the existence of Z(4430), a so-called "exotic hadron". Hadrons are composite particles made of quarks (and held together by gluons). According to the quark model, hadrons can only form in one of two ways: a quark-antiquark pairing (known as a meson) and in a quark triplet (know as a baryon).
The most common hadrons in the universe are protons and neutrons which form atomic nuclei. Protons consist of 2 up-type quarks and 1 down-type quark ("uud") while neutrons consist of 1 up quark and 2 down quarks ("udd"). The names and details of quark types, which are known as "flavours", is not something I will go into here, as it is an interesting enough topic to deserve its own post (although a proper explanation for the layperson would need a little more than that I think).
The quark model is a simple one though and does not describe all of the dynamics permitted by quantum chromodynamics ("QCD", the part of the Standard Model that describes strong interactions). This leaves open the door for exotic hadrons which are not mesons or baryons. Z(4430) is one such exotic hadron.
It was first 'discovered' in 2007 (although 5-sigma confirmation didn't come until 2008) and has now been observed at the LHCb experiment at 13.9-sigma accuracy. This means the chances of the observation being a statistical fluke are $1$ in $1.579\times10^{43}$ (a very, very, very large number indeed). It is believed to be a tetraquark made up of 1 charm quark, 1 charm antiquark, 1 down quark and 1 up antiquark ("ccdu").
While perhaps not as exciting as, for example, the BICEP2 result recently, this confirmation is still a very interesting result and will hopefully spur on further developments in the search for exotic hadrons.
Thursday, April 3, 2014
Learning to learn
It has happened a few times over the past couple of years that someone will come up to me with some piece of mathematics that they're having trouble with and ask me if I can help them with it. I'll have a look over it and say something along the lines of "I'm not familiar with this specifically, but I think I should be able to pick it up pretty easily". The usual response to this involves some degree of indignation.
Of course I understand why. Nobody likes to feel stupid or belittled, and this is certainly not my intention, but I can see how someone might feel that way. If I were in that position it would probably be my first response as well. But with that said, I don't think it's an entirely reasonable one.
I'm not trying to show off how clever I am. They were the ones to come to me asking for my help (and if I can I try my best to). Presumably they came to me because they expected I had some degree of expertise, but I get the impression the sort of expertise they were expecting was that of having been taught the subject previously. Sometimes this is the case of course, but mathematics is such a broad topic that you will rarely have seen everything at a sub-postgraduate level no matter how educated you are.
The point of having gone to university and graduated with a degree (even a bachelor's) specialising in mathematics or a mathematics-heavy field like physics is not to simply learn as much of a field as you can and leave it there. The point is to equip yourself with the tools to teach yourself new things, and I hope to some extent I have managed to do that. I've taught myself things I never saw at university and so I know my limitations. Some concepts and techniques I'm reasonably familiar with and others I'm not; some things take more time and more effort to learn than others. It just happens that through study and experience I've managed to achieve the minimal level of competence to teach myself what they're having trouble with.
So my exasperation does not come from a place of ignorance. I have struggled with maths in the past, I know what it's like. The fact remains, however, that my degree would be worth very little if I hadn't picked up the ability to learn high school- or early undergraduate-level maths with relative ease. If someone's problem left me just as nonplussed as they were, then I should probably be seeking partial reimbursement from my university. I certainly hope they don't take it as a personal slight; I have no doubt that if they had studied what I had then they would have just as little trouble with their present difficult as I might (and, I would hazard, quite possibly even less so).
Of course I understand why. Nobody likes to feel stupid or belittled, and this is certainly not my intention, but I can see how someone might feel that way. If I were in that position it would probably be my first response as well. But with that said, I don't think it's an entirely reasonable one.
I'm not trying to show off how clever I am. They were the ones to come to me asking for my help (and if I can I try my best to). Presumably they came to me because they expected I had some degree of expertise, but I get the impression the sort of expertise they were expecting was that of having been taught the subject previously. Sometimes this is the case of course, but mathematics is such a broad topic that you will rarely have seen everything at a sub-postgraduate level no matter how educated you are.
The point of having gone to university and graduated with a degree (even a bachelor's) specialising in mathematics or a mathematics-heavy field like physics is not to simply learn as much of a field as you can and leave it there. The point is to equip yourself with the tools to teach yourself new things, and I hope to some extent I have managed to do that. I've taught myself things I never saw at university and so I know my limitations. Some concepts and techniques I'm reasonably familiar with and others I'm not; some things take more time and more effort to learn than others. It just happens that through study and experience I've managed to achieve the minimal level of competence to teach myself what they're having trouble with.
So my exasperation does not come from a place of ignorance. I have struggled with maths in the past, I know what it's like. The fact remains, however, that my degree would be worth very little if I hadn't picked up the ability to learn high school- or early undergraduate-level maths with relative ease. If someone's problem left me just as nonplussed as they were, then I should probably be seeking partial reimbursement from my university. I certainly hope they don't take it as a personal slight; I have no doubt that if they had studied what I had then they would have just as little trouble with their present difficult as I might (and, I would hazard, quite possibly even less so).
Sunday, March 30, 2014
A very dangerous factory
Suppose a new type of bomb is invented whose detonation device is so incredibly sensitive that if it comes into contact with a single particle it will explode. Putting aside the impracticality of such a weapon (and the obvious factory OH&S issues), the producer wishes to maintain quality control as, with anything, some bombs will be faulty and not have detonation devices attached. The question immediately arises: Is it possible to have some ensemble of bombs which we can guarantee contains no faulty weapons?
This question is known as the Elitzur-Vaidman bomb-testing problem, and although one can arrive after reasonably little thought at the fairly obvious answer that no such ensemble is possible (as any direct observation using light or matter will detonate any working bombs), in actual fact such an ensemble is possible! How can this be the case? The short answer is: quantum effects. The long answer? Read on!
The solution to this problem involves the use of a Mach-Zehnder interferometer (Fig. 1) with a single-photon source. To see how, let's consider the case of the interferometer without any bomb in place. We then have
\begin{align}\label{eq:MZ}
\left|s\right\rangle &\rightarrow \frac{i}{\sqrt{2}}\left|u\right\rangle + \frac{1}{\sqrt{2}}\left|v\right\rangle \nonumber \\
&\rightarrow \frac{i}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}\left|c\right\rangle + \frac{i}{\sqrt{2}}\left|d\right\rangle\right) + \frac{1}{\sqrt{2}}\left(\frac{i}{\sqrt{2}}\left|c\right\rangle + \frac{1}{\sqrt{2}}\left|d\right\rangle\right) \nonumber \\
&= \frac{i}{2}\left|c\right\rangle + \frac{-1}{2}\left|d\right\rangle + \frac{i}{2}\left|c\right\rangle + \frac{1}{2}\left|d\right\rangle \nonumber \\
&= i\left|c\right\rangle,
\end{align}
where $\left|a\right\rangle$ represents the quantum state in the $a$-branch of the interferometer (as labelled in Fig. 1) and $i$ is the imaginary unit.$^{1}$ What the above calculation shows$^{2}$ is that (somewhat surprisingly) despite the branching at the second beam-splitter, destructive interference along $d$ and constructive interference along $c$ causes the photon to always be detected at $C$ and never at $D$ (for this alignment).
Now let's consider the same Mach-Zehnder interferometer but with a bomb placed such that the detector will be along the $u$-branch (as shown in Fig. 2). In this case we have
\begin{align}\label{eq:bomb}
\left|s\right\rangle\left|B_0\right\rangle &\rightarrow \frac{i}{\sqrt{2}}\left|u\right\rangle\left|B_0\right\rangle + \frac{1}{\sqrt{2}}\left|v\right\rangle\left|B_0\right\rangle \nonumber \\
&\rightarrow \frac{i}{\sqrt{2}}\left|X\right\rangle + \frac{1}{\sqrt{2}}\left|v\right\rangle\left|B_0\right\rangle \nonumber \\
&\rightarrow \frac{i}{\sqrt{2}}\left|X\right\rangle + \frac{1}{\sqrt{2}}\left(\frac{i}{\sqrt{2}}\left|c\right\rangle + \frac{1}{\sqrt{2}}\left|d\right\rangle\right)\left|B_0\right\rangle \nonumber \\
&= \frac{i}{\sqrt{2}}\left|X\right\rangle +\frac{i}{2}\left|c\right\rangle\left|B_0\right\rangle + \frac{1}{2}\left|d\right\rangle\left|B_0\right\rangle,
\end{align}
where $\left|B_0\right\rangle$ is the 'primed' or unexploded bomb, $\left|X\right\rangle$ represents the state where the bomb has been detonated$^3$ and $\left|a\right\rangle\left|b\right\rangle\equiv\left|a\right\rangle\otimes\left|b\right\rangle$. Note that for the purposes of this thought experiment we are assuming the detonator is a perfect detector, i.e., the photon wave cannot travel down $u$ without being absorbed.
As is clear from equation \ref{eq:bomb}, the inclusion of the detonator destroys the constructive/destructive interference that caused the simplification in equation \ref{eq:MZ}. Therefore, in the detonator case, rather than having every photon detected at $C$, we have the photon detected at $C$ with a probability of $1/4$, detected at $D$ with a probability of $1/4$ and the bomb detonated with a probability of $1/2$.$^4$
This is what makes it possible to assemble a set of functional bombs without detonating them—if a photon is detected by $D$ then the bomb must have a detonator attached and so we can set it aside knowing it works. If a photon is detected by $C$ then the functionality is indeterminate as we expect a detection at $C$ with non-zero probability in both detonator and no-detonator cases, but this is not a problem as we can simply emit another photon and re-run the test.
Note that while the probabilities above can be derived (in a fairly straightforward manner) from classical principles, we cannot apply a classical interpretation here as the quantum nature of the experiment is indispensable. In the classical (many-photon) run it is possible to both detonate a bomb and make a detection at $D$; this is precluded in the quantum case as the single photon cannot be absorbed by multiple objects. Furthermore, it is the wave-nature of the photon that permits the destructive interference at $D$ in the no-detonator case and thus provides 'detection by $D$' to signify the presence of the detonator and thus successfully make an 'interaction-free' measurement.
If you're unconvinced of this argument because it is based on a purely theoretical consideration, consider that this thought experiment has (equivalently) been carried out in the real world (admittedly using an ordinary detector rather than a bomb) and in fact was first done about a year after this problem was first published. I can't speak to the practical applications, if any exist, but I love this problem regardless for the simple fact that the solution challenges your intuition but can be understood using reasonably straightforward quantum mechanical principles.
Consider a beam-splitter as shown in Fig. 3. This system can be represented by the matrix equation $\left|\psi_3,\psi_4\right\rangle = \hat{B}\left|\psi_1,\psi_2\right\rangle$, or explicitly,
\begin{equation}\label{eq:BSM}
\begin{pmatrix}
\psi_3 \\ \psi_4
\end{pmatrix}
=
\begin{pmatrix}
T & R \\ R & T
\end{pmatrix}
\begin{pmatrix}
\psi_1 \\ \psi_2
\end{pmatrix},
\end{equation}
where $T$ and $R$ are the transmission and reflection coefficients respectively. In the experiment we assume an ideal, lossless beam-splitter which demands that the beam-splitter matrix be unitary, i.e., $\hat{B}^{\dagger}\hat{B}=\hat{\mathbb{I}}$, or,
\begin{equation}\label{eq:unitary}
\begin{pmatrix}
T^{\ast} & R^{\ast} \\ R^{\ast} & T^{\ast}
\end{pmatrix}
\begin{pmatrix}
T & R \\ R & T
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix}.
\end{equation}
Equation \ref{eq:unitary} immediately implies the following relations:
\begin{equation}
|T|^2+|R|^2=1,
\end{equation}
\begin{equation}\label{eq:0}
T^{\ast}R+R^{\ast}T=0.
\end{equation}
As $T$ and $R$ are complex numbers, we can represent them in polar form as $T=|T|e^{i\theta_T}$ and $R=|R|e^{i\theta_R}$. For simplicity we choose $\theta_T=0$ and thus $T=|T|\implies T^{\ast}=T$ and so equation \ref{eq:0} becomes
\begin{align}\label{eq:0new}
T|R|e^{i\theta_R}+|R|e^{-i\theta_R}T&=0 \nonumber \\
2T|R|\cos{\left(\theta_R\right)}&=0
\end{align}
where we have made use of the identity $\cos{(\alpha)}=e^{i\alpha}/2+e^{-i\alpha}/2$. Equation \ref{eq:0new} is satisfied by $\theta_R=n\pi+\pi/2, n\in\mathbb{Z}$, but we will choose $n=0\implies\theta_R=\pi/2$ for simplicity, which in turn gives $R=|R|e^{i\pi/2}=i|R|$.
Finally, as the beam-splitter is 50:50 (50% transmission, 50% reflection) we demand $|T|=|R|=1/\sqrt{2}$ and so the beam-splitter matrix is given by
\begin{equation}\label{eq:B}
\hat{B}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & i \\ i & 1
\end{pmatrix}.
\end{equation}
It should be clear that equation \ref{eq:B} is not a unique representation of $\hat{B}$; another choice of $\theta_T$ and/or $\theta_R$ would yield a different (unitary) matrix that would make no difference to the calculations shown in equations \ref{eq:MZ} and \ref{eq:bomb} (I leave proof of this as an exercise for the interested reader). With that said, the reason I like this representation is that it allows $i$ to function as a label for the states that result from a beam-splitter reflection, making it easier to write down interferometer equations directly from the diagram and keep track of where each term comes from. This is, of course, purely a matter of personal preference.
$2$. This equation is an example of quantum superposition in action. For example, the first line says that the photon exists in a superposition of the $\left|u\right\rangle$ and $\left|v\right\rangle$ states where the states are equally weighted (as we are assuming normalisation). Superposition is a fundamental aspect of quantum mechanics that follows from the linearity of the Schrödinger equation (linear combinations of solutions will themselves be solutions). In this case, the beam-splitter splits the photon probability wave along the two channels and so in some sense the photon travels along both branches, although no measurement can be made which will detect the photon in both channels at once—this is not a consequence of experimental limitations but is a restriction that is fundamental to quantum theory. The question of why this is the case is a deep and ongoing one, and I encourage the interested reader to investigate the literature on the philosophy (and especially interpretations) of quantum mechanics.
$3$. I have gone to some pains in this post to avoid using the term "wavefunction collapse" at any point, although for clarity I will say will say that in the Copenhagen interpretation, the case of the photon interacting with the detonator (or any of the detectors for that matter) is an example of wavefunction collapse.
$4$. So long as the beam-splitters are both 50:50, as we have assumed throughout this blog post. Naturally, other types of beam-splitters will yield different results, and in fact using a more sophisticated apparatus will permit a much better detection level (in theory, the detection fraction can be brought arbitrarily close to 1, although I cannot speak to the practicality of such an apparatus).
This question is known as the Elitzur-Vaidman bomb-testing problem, and although one can arrive after reasonably little thought at the fairly obvious answer that no such ensemble is possible (as any direct observation using light or matter will detonate any working bombs), in actual fact such an ensemble is possible! How can this be the case? The short answer is: quantum effects. The long answer? Read on!
 |
| Figure 1: A Mach-Zehnder interferometer with faulty bomb $B$ in place and all branches labelled. Note that the $d$-branch is drawn only for illustrative purposes; the photon cannot be detected along the $d$-branch due to destructive interference (see equation \ref{eq:MZ}). (1) The single-photon source $S$. (2) One of the the two 50:50 beam-splitters which are both assumed to be lossless. (3) One of the two mirrors which are assumed to be perfectly reflective. (4) One of the two detectors which are assumed to be perfect detectors. |
The solution to this problem involves the use of a Mach-Zehnder interferometer (Fig. 1) with a single-photon source. To see how, let's consider the case of the interferometer without any bomb in place. We then have
\begin{align}\label{eq:MZ}
\left|s\right\rangle &\rightarrow \frac{i}{\sqrt{2}}\left|u\right\rangle + \frac{1}{\sqrt{2}}\left|v\right\rangle \nonumber \\
&\rightarrow \frac{i}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}\left|c\right\rangle + \frac{i}{\sqrt{2}}\left|d\right\rangle\right) + \frac{1}{\sqrt{2}}\left(\frac{i}{\sqrt{2}}\left|c\right\rangle + \frac{1}{\sqrt{2}}\left|d\right\rangle\right) \nonumber \\
&= \frac{i}{2}\left|c\right\rangle + \frac{-1}{2}\left|d\right\rangle + \frac{i}{2}\left|c\right\rangle + \frac{1}{2}\left|d\right\rangle \nonumber \\
&= i\left|c\right\rangle,
\end{align}
where $\left|a\right\rangle$ represents the quantum state in the $a$-branch of the interferometer (as labelled in Fig. 1) and $i$ is the imaginary unit.$^{1}$ What the above calculation shows$^{2}$ is that (somewhat surprisingly) despite the branching at the second beam-splitter, destructive interference along $d$ and constructive interference along $c$ causes the photon to always be detected at $C$ and never at $D$ (for this alignment).
.png) |
| Figure 2: A Mach-Zehnder interferometer with working bomb $B$ in place and all branches labelled. Note that $B$ blocks the $u$-branch whether the photon interacts with the detector or not (the case of an interaction is not illustrated here as this would correspond to the detonation of the bomb). |
Now let's consider the same Mach-Zehnder interferometer but with a bomb placed such that the detector will be along the $u$-branch (as shown in Fig. 2). In this case we have
\begin{align}\label{eq:bomb}
\left|s\right\rangle\left|B_0\right\rangle &\rightarrow \frac{i}{\sqrt{2}}\left|u\right\rangle\left|B_0\right\rangle + \frac{1}{\sqrt{2}}\left|v\right\rangle\left|B_0\right\rangle \nonumber \\
&\rightarrow \frac{i}{\sqrt{2}}\left|X\right\rangle + \frac{1}{\sqrt{2}}\left|v\right\rangle\left|B_0\right\rangle \nonumber \\
&\rightarrow \frac{i}{\sqrt{2}}\left|X\right\rangle + \frac{1}{\sqrt{2}}\left(\frac{i}{\sqrt{2}}\left|c\right\rangle + \frac{1}{\sqrt{2}}\left|d\right\rangle\right)\left|B_0\right\rangle \nonumber \\
&= \frac{i}{\sqrt{2}}\left|X\right\rangle +\frac{i}{2}\left|c\right\rangle\left|B_0\right\rangle + \frac{1}{2}\left|d\right\rangle\left|B_0\right\rangle,
\end{align}
where $\left|B_0\right\rangle$ is the 'primed' or unexploded bomb, $\left|X\right\rangle$ represents the state where the bomb has been detonated$^3$ and $\left|a\right\rangle\left|b\right\rangle\equiv\left|a\right\rangle\otimes\left|b\right\rangle$. Note that for the purposes of this thought experiment we are assuming the detonator is a perfect detector, i.e., the photon wave cannot travel down $u$ without being absorbed.
As is clear from equation \ref{eq:bomb}, the inclusion of the detonator destroys the constructive/destructive interference that caused the simplification in equation \ref{eq:MZ}. Therefore, in the detonator case, rather than having every photon detected at $C$, we have the photon detected at $C$ with a probability of $1/4$, detected at $D$ with a probability of $1/4$ and the bomb detonated with a probability of $1/2$.$^4$
This is what makes it possible to assemble a set of functional bombs without detonating them—if a photon is detected by $D$ then the bomb must have a detonator attached and so we can set it aside knowing it works. If a photon is detected by $C$ then the functionality is indeterminate as we expect a detection at $C$ with non-zero probability in both detonator and no-detonator cases, but this is not a problem as we can simply emit another photon and re-run the test.
Note that while the probabilities above can be derived (in a fairly straightforward manner) from classical principles, we cannot apply a classical interpretation here as the quantum nature of the experiment is indispensable. In the classical (many-photon) run it is possible to both detonate a bomb and make a detection at $D$; this is precluded in the quantum case as the single photon cannot be absorbed by multiple objects. Furthermore, it is the wave-nature of the photon that permits the destructive interference at $D$ in the no-detonator case and thus provides 'detection by $D$' to signify the presence of the detonator and thus successfully make an 'interaction-free' measurement.
If you're unconvinced of this argument because it is based on a purely theoretical consideration, consider that this thought experiment has (equivalently) been carried out in the real world (admittedly using an ordinary detector rather than a bomb) and in fact was first done about a year after this problem was first published. I can't speak to the practical applications, if any exist, but I love this problem regardless for the simple fact that the solution challenges your intuition but can be understood using reasonably straightforward quantum mechanical principles.
Notes
$1$. The inclusion of $i$ in these equations might seem unusual or arbitrary, so I will provide a derivation here that shows where it comes from. |
| Figure 3: A beam-splitter with two incoming beams ($\psi_1$ and $\psi_2$) and two outgoing beams ($\psi_3$ and $\psi_4$). The incoming and outgoing beams are related by the beam-splitter matrix for the beam-splitter in question, as shown in equation \ref{eq:BSM}. In the note below, the beam-splitter will be assumed to be 50:50 in accordance with the calculations in the main text. |
Consider a beam-splitter as shown in Fig. 3. This system can be represented by the matrix equation $\left|\psi_3,\psi_4\right\rangle = \hat{B}\left|\psi_1,\psi_2\right\rangle$, or explicitly,
\begin{equation}\label{eq:BSM}
\begin{pmatrix}
\psi_3 \\ \psi_4
\end{pmatrix}
=
\begin{pmatrix}
T & R \\ R & T
\end{pmatrix}
\begin{pmatrix}
\psi_1 \\ \psi_2
\end{pmatrix},
\end{equation}
where $T$ and $R$ are the transmission and reflection coefficients respectively. In the experiment we assume an ideal, lossless beam-splitter which demands that the beam-splitter matrix be unitary, i.e., $\hat{B}^{\dagger}\hat{B}=\hat{\mathbb{I}}$, or,
\begin{equation}\label{eq:unitary}
\begin{pmatrix}
T^{\ast} & R^{\ast} \\ R^{\ast} & T^{\ast}
\end{pmatrix}
\begin{pmatrix}
T & R \\ R & T
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix}.
\end{equation}
Equation \ref{eq:unitary} immediately implies the following relations:
\begin{equation}
|T|^2+|R|^2=1,
\end{equation}
\begin{equation}\label{eq:0}
T^{\ast}R+R^{\ast}T=0.
\end{equation}
As $T$ and $R$ are complex numbers, we can represent them in polar form as $T=|T|e^{i\theta_T}$ and $R=|R|e^{i\theta_R}$. For simplicity we choose $\theta_T=0$ and thus $T=|T|\implies T^{\ast}=T$ and so equation \ref{eq:0} becomes
\begin{align}\label{eq:0new}
T|R|e^{i\theta_R}+|R|e^{-i\theta_R}T&=0 \nonumber \\
2T|R|\cos{\left(\theta_R\right)}&=0
\end{align}
where we have made use of the identity $\cos{(\alpha)}=e^{i\alpha}/2+e^{-i\alpha}/2$. Equation \ref{eq:0new} is satisfied by $\theta_R=n\pi+\pi/2, n\in\mathbb{Z}$, but we will choose $n=0\implies\theta_R=\pi/2$ for simplicity, which in turn gives $R=|R|e^{i\pi/2}=i|R|$.
Finally, as the beam-splitter is 50:50 (50% transmission, 50% reflection) we demand $|T|=|R|=1/\sqrt{2}$ and so the beam-splitter matrix is given by
\begin{equation}\label{eq:B}
\hat{B}=\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & i \\ i & 1
\end{pmatrix}.
\end{equation}
It should be clear that equation \ref{eq:B} is not a unique representation of $\hat{B}$; another choice of $\theta_T$ and/or $\theta_R$ would yield a different (unitary) matrix that would make no difference to the calculations shown in equations \ref{eq:MZ} and \ref{eq:bomb} (I leave proof of this as an exercise for the interested reader). With that said, the reason I like this representation is that it allows $i$ to function as a label for the states that result from a beam-splitter reflection, making it easier to write down interferometer equations directly from the diagram and keep track of where each term comes from. This is, of course, purely a matter of personal preference.
$2$. This equation is an example of quantum superposition in action. For example, the first line says that the photon exists in a superposition of the $\left|u\right\rangle$ and $\left|v\right\rangle$ states where the states are equally weighted (as we are assuming normalisation). Superposition is a fundamental aspect of quantum mechanics that follows from the linearity of the Schrödinger equation (linear combinations of solutions will themselves be solutions). In this case, the beam-splitter splits the photon probability wave along the two channels and so in some sense the photon travels along both branches, although no measurement can be made which will detect the photon in both channels at once—this is not a consequence of experimental limitations but is a restriction that is fundamental to quantum theory. The question of why this is the case is a deep and ongoing one, and I encourage the interested reader to investigate the literature on the philosophy (and especially interpretations) of quantum mechanics.
$3$. I have gone to some pains in this post to avoid using the term "wavefunction collapse" at any point, although for clarity I will say will say that in the Copenhagen interpretation, the case of the photon interacting with the detonator (or any of the detectors for that matter) is an example of wavefunction collapse.
$4$. So long as the beam-splitters are both 50:50, as we have assumed throughout this blog post. Naturally, other types of beam-splitters will yield different results, and in fact using a more sophisticated apparatus will permit a much better detection level (in theory, the detection fraction can be brought arbitrarily close to 1, although I cannot speak to the practicality of such an apparatus).
Thursday, March 20, 2014
News (2014/03/20)
I'm trying to post on my blog much more often this year than I used to, in fact as close to every week as I can manage. Unfortunately, the post I'm working on at the moment isn't nearly ready for publication, so this week I'm instead going to make a little news post, the first item of which will be the thing that I just told you (about the new blog post coming soon)!
The next item of news is not particularly new; last Friday (the 14th of March) was Pi Day. Pi Day is of course silly for a whole bunch of reasons (the main one being it's based on the completely nonsensical American dating system) so I've decided to introduce readers who may not be familiar with it to the concept of tau (τ). Tau has been proposed as an alternative to pi, and while I am not especially partisan on the matter I have to say I am somewhat sympathetic. Here is the case for tau laid out in the Tau Manifesto and for the sake of fairness a counterargument in the Pi Manifesto.
Finally, the real news comes in the form of the results of the recent BICEP2 measurements of B-mode polarisation in the CMB, easily the biggest news in physics since the Higgs was announced in 2012 and a major breakthrough for early-universe cosmologists. I will be able to do an explanatory post about the news if there's enough demand for one, but otherwise a lot of good explanations can be found around the place ranging from the somewhat simplistic to the slightly more technical. This is a very exciting time for fundamental physics and I expect to see some very interesting papers published in the next couple of years based on insights from this new data.
Finally, it isn't technically news, but if you haven't heard of them already, I strongly urge you to check out Brady Haran's science channels, especially Sixty Symbols (physics) and Numberphile (mathematics); I've subscribed to most of them on YouTube and they are absolutely fantastic.
That's all for this quick post, hopefully I'll have a considerably more in-depth number ready for next week! See you then!
The next item of news is not particularly new; last Friday (the 14th of March) was Pi Day. Pi Day is of course silly for a whole bunch of reasons (the main one being it's based on the completely nonsensical American dating system) so I've decided to introduce readers who may not be familiar with it to the concept of tau (τ). Tau has been proposed as an alternative to pi, and while I am not especially partisan on the matter I have to say I am somewhat sympathetic. Here is the case for tau laid out in the Tau Manifesto and for the sake of fairness a counterargument in the Pi Manifesto.
Finally, the real news comes in the form of the results of the recent BICEP2 measurements of B-mode polarisation in the CMB, easily the biggest news in physics since the Higgs was announced in 2012 and a major breakthrough for early-universe cosmologists. I will be able to do an explanatory post about the news if there's enough demand for one, but otherwise a lot of good explanations can be found around the place ranging from the somewhat simplistic to the slightly more technical. This is a very exciting time for fundamental physics and I expect to see some very interesting papers published in the next couple of years based on insights from this new data.
Finally, it isn't technically news, but if you haven't heard of them already, I strongly urge you to check out Brady Haran's science channels, especially Sixty Symbols (physics) and Numberphile (mathematics); I've subscribed to most of them on YouTube and they are absolutely fantastic.
That's all for this quick post, hopefully I'll have a considerably more in-depth number ready for next week! See you then!
Thursday, March 6, 2014
Playing with infinite series
Sequences, series and summation notation
In mathematical parlance, the term 'sequence' carries a similar meaning as in regular speech; it refers to an ordered list of numbers that usually follows a rule. For example, the sequence $1, 2, 3, 4, 5...$ (onwards without end) is given by adding 1 to the previous number in the sequence, beginning at 1. The definition of the term 'series' is less obvious—one description is that a series is the sum of a sequence, or in other words, to gain a series one adds all the terms of a sequence. For a finite series (a series with finitely many terms, or alternatively a series with a last term), the mathematics is typically fairly straightforward and so we will focus instead on the topic of infinite series, but before we do so, I will make a brief digression to discuss notation.For reasons I am sympathetic to, few people enjoy reading about mathematical notation and it is difficult to write about it in a way which is interesting. However, while it is possible to write a series as, for example, $1+2+3+4+5+...$ this soon becomes cumbersome and is very limiting for finite series with a large number of terms and for series for which the rule or pattern is not obvious from the first few terms. For these reasons, mathematicians have developed a notation for series which I will adopt in the remainder of this post for instructional purposes (side–by–side with the long-hand version for clarity).
The notation is referred to as summation notation or sigma notation and uses a large capital sigma $\Sigma$ to denote the summation. The rule is written to the right of the sigma in terms of an index of summation, the lowest value of the index will be under the sigma and the highest value will be above the sigma (or '$\infty$' for an infinite series with no final term).$^1$
A simple example of sigma notation in action is the finite series
\begin{align} \label{eq:square5}
\sum_{n=1}^{5}n^2 & = 1^2+2^2+3^2+4^2+5^2 \nonumber \\
& = 1+4+9+16+25,
\end{align}
which in this case is equal to $55$. Here, the index of summation $n$ appears prominently in the rule as the number that is being squared. Another example, slightly tricker this time, is the infinite series
\begin{align} \label{eq:Zeno}
\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n
& = \left(\frac{1}{2}\right)^1+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+\left(\frac{1}{2}\right)^4+... \nonumber \\ & = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...
\end{align}
where the index of summation is this time the power that $1/2$ is raised to in each term of the sum.$^2$
Infinite series behaving badly
An infinite series is in many ways a different beast to a finite series. Possibly the clearest way is conceptually; finite series are computed fairly easily, in principle at least—it is simply a case of adding so many numbers together and then reading the value off your calculator. This is not possible with an infinite series, as there is no final term and no opportunity to hit a final '$=$' button on the calculator, as the sum goes on forever. This is not always a problem, however. Some series are known as 'convergent', which is to say that they are in effect equal or equivalent to some number. There are many tests for convergence, but suffice it to say that we know a series is convergent when it approaches said number.$^3$ We have already seen a convergent series in equation \ref{eq:Zeno}, but another example of a convergent series is\begin{equation}
\label{eq:euler}
e=\sum^{\infty}_{n=0}\frac{1}{n!}=1+1+\frac{1}{2}+\frac{1}{6}+...
\end{equation}
where $!$ is the factorial operator.$^4$ As you can see, this series converges to $e$, a truly marvellous number which happens to be my favourite mathematical constant, but an explanation of that fact would require another post entirely.
If a series is not convergent, however, then we call it divergent,$^5$ and that is where the trouble starts. A straightforward example is the harmonic series
\begin{equation}
\label{eq:harmonic}
\sum^{\infty}_{n=1}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...
\end{equation}
which despite its similarity to equation \ref{eq:Zeno} increases indefinitely and does not approach any particular number. This is in some sense easy to understand intuitively; one can see how the sum would just keep getting bigger and bigger. This kind of reasoning is based around partial sums, which are truncations of infinite series. For the harmonic series the partial sum is given by
\begin{equation} \label{eq:Hn}
H_n=\sum^{n}_{k=1}\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n},
\end{equation}
where the $n^{\text{th}}$ partial sum $H_n$ is known as the $n^{\text{th}}$ harmonic number. (Note that here we use $n$ to label the partial sum and so the role of the index of summation is taken over by $k$ to avoid confusion, although we could have chosen $k$ as our label and kept $n$ the index of summation if we wished; it makes no difference). Partial sums are finite series by design and so just like equation \ref{eq:square5} there is a final term after which we can hit the metaphorical '$=$' button. If we do so after 2 terms we find $H_2=3/2$, after 3 terms $H_3=11/6$, 4 terms $H_4=25/12$ and so on.$^6$
Now let us consider an altogether different beast known as Grandi's series. Grandi's series is given as
\begin{equation} \label{eq:Grandi}
\sum^{\infty}_{n=0}(-1)^n=1-1+1-1+1-1...
\end{equation}
Unlike the harmonic series, the partial sums of Grandi's series do not seem to trend in any particular direction, but rather alternate between two 'accumulation points' at 1 and 0. This peculiarity will cause us some trouble, but to see why first we will make a brief foray into the physical sciences.
A brief foray into the physical sciences
Suppose we have two thin neutral conductive plates placed parallel to each other very close together in a vacuum. According to classical physics (and everyday intuition) absolutely nothing will happen. However, this is not what we observe; in fact, the two plates will experience an attractive force attempting to bring them together. This is known as the Casimir effect, and can only be understood in terms of quantum mechanics. The Casimir effect is typically expressed in terms of quantum electrodynamics, but there is nothing inherently electrodynamic about the effect and so one can consider many analogous scenarios with equivalent Casimir effects. In order to greatly simplify our derivation I will choose to do just that.One thing that it is essential to be aware of is that in quantum mechanics a vacuum is not 'empty' in the sense that there is nothing at all there; so far as we understand, such an emptiness cannot exist in the physical universe. This possibility is precluded both on experimental grounds and on theoretical grounds by the uncertainty principle.$^7$ Instead we understand the universe to be filled with quantised fields (mathematically speaking, a field assigns some value(s) to every point in space(time); a quantised field is one where the range of possible values is restricted to some discrete set) with each particle being a localised excitation in the energy of the field, e.g., photons (light particles) are excitations in the electromagnetic field, electrons are excitations in the electron field and so on. The rough procedure$^8$ for quantising a field is to treat it as a quantum harmonic oscillator (QHO) at every point in space—one could crudely picture this as an infinite system of connected balls and springs—which naturally results in quantised, discrete energy levels. A 'true' vacuum is therefore the ground state (lowest energy state) in every quantum field. We know that the ground state energy cannot be 0 as that would be the kind of emptiness that cannot exist. Rather, it takes on a value of
\begin{equation} \label{eq:energy}
E_0=\frac{\hbar\omega}{2},
\end{equation}
the ground state of the QHO, where $\hbar$ is the reduced Planck constant and $\omega$ is the angular frequency of the oscillator.
With all this in mind, let's return to the Casimir effect, considering a 1+1-dimensional massless scalar field to simplify and clarify the example. The plates impose what are called 'boundary conditions', they restrict the frequencies that waves in between the plates can take, in this case to standing waves.$^9$ The equation for a standing wave in 1+1-dimensions is
\begin{equation} \label{eq:wave}
\psi_n(x,t)=e^{-i\omega_nt}\sin{\left(\frac{n\pi x}{a}\right)}
\end{equation}
where $a$ is the width of the cavity between the plates, $n$ is a natural number (a positive whole number) and $\omega_n$ is the angular frequency given by
\begin{equation} \label{eq:omega}
\omega_n=\frac{n\pi c}{a}
\end{equation}
where $c$ is the wave speed. Now, we know that the ground state energy for a QHO is given by equation \ref{eq:energy} and we know that in between the plates only standing waves can exist, so we can only have waves with angular frequency $\omega_n=n\pi c/a$. If we wish to find the vacuum energy between the plates, it seems clear then that all we need to do is sum over the possible ground state energies, giving
\begin{equation} \label{eq:vacuum}
E=\frac{\hbar}{2}\sum^{\infty}_{n=1}\omega_n=\frac{\hbar\pi c}{2a}\sum^{\infty}_{n=1}n.
\end{equation}
Here we run into a problem. Unless you haven't been paying attention, you'll notice that
\begin{equation} \label{eq:natural}
\sum^{\infty}_{n=1}n=1+2+3+4+...
\end{equation}
absolutely positively does not converge at all, and yet here it appears in a physics equation relating to a very real and decidedly measurably finite effect. How can this be? We haven't made a mistake, but we have overlooked one crucial fact. No matter what our plates are made of, they cannot confine arbitrarily high energies of the field; those high energy modes will always be able to escape. So what we need now is to somehow take account of that fact and in doing so somehow assign a finite value to equation \ref{eq:vacuum} and rescue our derivation.
Putting divergent series to work
So what we seek is a way of attaching a meaningful finite value to divergent equations. Let's take a look at Grandi's series again and see if we can come up with anything consistent. We can try cancelling off pairs of terms to give\begin{align} \label{eq:gpair1}
\sum^{\infty}_{n=0}(-1)^n&=(1-1)+(1-1)+(1-1)+...\nonumber\\
&=0+0+0+...\nonumber\\
&=0
\end{align}
but we can just as easily choose different pairings to give
\begin{align} \label{eq:gpair2}
\sum^{\infty}_{n=0}(-1)^n&=1+(-1+1)+(-1+1)+...\nonumber\\
&=1+0+0+...\nonumber\\
&=1
\end{align}
which is certainly not consistent. We can try re-ordering the series to bring all the $+1$s to the front, but this gives
\begin{align} \label{eq:inf+1}
\sum^{\infty}_{n=0}(-1)^n=1+1+1+...-1-1-1...
\end{align}
As there are an infinite number of $+1$s we never reach the $-1$s and the series approaches $+\infty$. Trying the same process by arranging all the $-1$s to the front will in the same way cause the series to approach $-\infty$. Rather than find a single consistent way of assigning a number to the series, all we have found is four duds.
The reason these methods are all duds is because operations like reordering and cancelling pairs of terms (method of differences) are valid only for convergent series. If we try to apply them to divergent series, the result is clearly a mess. Let's step back and look at the problem from another angle. We want to assign a number to the series; presumably we should be able to manipulate that number algebraically. If we call the series $S$ then after some algebraic juggling we find
\begin{align} \label{eq:S}
S&=1-1+1-1+1-1+...\nonumber\\
1-S&=1-(1-1+1-1+1-1+...)\nonumber\\
&=1-1+1-1+1-1+...\nonumber\\
&=S\nonumber\\
1&=2S\nonumber\\
\Rightarrow S&=1/2
\end{align}
Furthermore, we can consider Grandi's series as an example of the infinite geometric series
\begin{align} \label{eq:geometricG}
\sum_{k=0}^{\infty}ar^k=a+ar+ar^2+ar^3...
\end{align}
where $a=1$ and $r=-1$. Even though Grandi's series is divergent, equation \ref{eq:geometricG} is convergent for $|r|<1$ and in that case
\begin{align} \label{eq:geometricC}
\sum_{k=0}^{\infty}ar^k=\frac{a}{1-r}.
\end{align}
If we substitute $a=1$ and $r=-1$ into equation \ref{eq:geometricC} then we again find $S=1/2$. Neither of these constitute solid proof in and of themselves, but they are highly suggestive. The tool we are looking for is the Cesàro sum.$^{10}$ A series is Cesàro summable when the mean value of its partial sums tends to a given value. For convergent series the Cesàro sum will always equal the number the series converges to and the Cesàro sum is defined for many divergent series too, including Grandi's series. The partial sums of Grandi's series are $1,0,1,0,...$ and so the terms in the Cesàro sequence are $1, 1/2, 2/3,1/2,3/5,1/2,4/7,...$ which clearly converges to $1/2$ in the limit.$^{11}$ In some sense this is quite a satisfying result as our Cesàro sum lies exactly in between the two accumulation points, serving as a kind of average value.
Having fun with zeta function regularisation
The partial sums of equation \ref{eq:natural} are the triangular numbers $T_n$ (so named because they give the numbers of objects that can be arranged into equilateral triangles) $1, 3, 6, 10, 15,...$, so we calculate the Cesàro sequence of equation \ref{eq:natural} and find it goes $1, 2, 10/3, 5, 7,...$ Once again, just as we stop to bask in our moment of triumph, we find our job isn't quite yet done; equation \ref{eq:natural} is not Cesàro summable. We must find another, more sophisticated method for attaching a number to that series.Let us consider the series
\begin{align} \label{eq:dirichlet}
D(s)=\sum^{\infty}_{n=1}n^{-s}, \text{ Re}(s)>1
\end{align}
where $s$ is a complex number and $\text{Re}(s)$ denotes the real part of $s$. For $s=-1$ this series would be exactly the same as equation \ref{eq:natural}, but the series is not defined for $s=-1$.$^{12}$ However, we saw back in equation \ref{eq:geometricC} that on that occasion if we applied the convergent case equation to Grandi's series we got the result of $1/2$ that turned out to be right; perhaps we could do something similar here? As it would happen, we can, but first I would implore you to, in the great words of John Arnold, "Hold on to your butts".
In the domain $\text{Re}(s)>1$, $\zeta(s)=D(s)$ where $\zeta(s)$ is known as the Riemann zeta function.$^{13}$ Unlike $D(s)$, $\zeta(s)$ is defined over the entire complex plane and is known as an analytic continuation of $D(s)$. Analytic continuation is a wonderfully useful (and perplexing) tool of complex analysis whereby the domain of an analytic ('well-behaved') function can be extended. While this may not seem like a big deal, it raises the question as to whether or not a function can be continued arbitrarily; if our original function is only defined for some small domain and we wish to extend that domain, what is to stop us giving it such–and–such value in the extended domain instead of some other value? As it would happen, the identity theorem states (very roughly) that any two holomorphic functions (all complex analytic functions are holomorphic) that are equal to each other at some point in a given domain must be equal over the entire domain, and thus there is only one unique way to analytically continue a function.
We wish to know what $\zeta(-1)$ is so we can assign that value to $D(-1)$ through the magic of analytic continuation. For negative whole numbers $n<0$,
\begin{equation} \label{eq:negzeta} \zeta(n)=-\frac{B_{1-n}}{1-n}
\end{equation}
where $B_{1-n}$ is the '$1-n$'$^{\text{th}}$ Bernoulli number.$^{14}$ In the case of $n=-1$ we have
\begin{equation} \label{eq:-1zeta}
\zeta(-1)=-\frac{B_2}{2}=-\frac{1}{12}.
\end{equation}
Thus we can assign to the divergent series $1+2+3+4+...$ the value of $-1/12$. If this strikes you as strange or even suspicious then I applaud your scepticism; there is indeed something plainly odd about assigning the value of a small, fractional negative number to a series of ever-increasing positive whole numbers. This is not at all like the neat case of Grandi's series where we had our value lying neatly between the accumulation points. But before we throw our hands up in despair, recall our motivation for this investigation, the Casimir effect. What happens if we use our value of $-1/12$ there?
As it would happen, to do so is to use a technique known as zeta function regularisation. We replace a divergent series with a 'regulator' in the form of a zeta function (although other regulators exist, each with different strengths and weaknesses) and in doing so remove unphysical infinities from our theory. If we have regularised correctly, then by the time we have reached our final result, the regulator will have disappeared—it is nothing more than a 'trick' for calculating the correct value and so it should not still appear at the last step.
\begin{equation} \label{eq:zetanorm}
E=\frac{\hbar\pi c}{2a}\sum^{\infty}_{n=1}n=\frac{\hbar\pi c}{2a}\zeta(-1)=-\frac{\hbar\pi c}{24a}.
\end{equation}
The force between the two plates is given by the negative gradient of the energy:
\begin{equation} \label{eq:force}
F=-\frac{\partial E}{\partial a}=-\frac{\partial}{\partial a}\left(-\frac{\hbar\pi c}{24a}\right)=-\frac{\hbar\pi c}{24a^2}
\end{equation}
which, lo and behold, is exactly the right result.
Final thoughts
The prompt for my writing this lengthy explanation of infinite series, divergences, regularisation and so on was a minor flurry on the Internet a little while ago due to a somewhat dodgy derivation of the result $1+2+3+4+...=-1/12$. In order to get this result a number of those forbidden–for–divergent–series operations were used for simplicity's sake, but in doing so I felt the important subtlety between a convergent series equalling a number and a divergent series being assigned a value (albeit in a rigorous way) was lost, and that is an important distinction to make. You cannot keep adding $1+2+3+4+...$ and then through the magic of infinity come up with a $-1/12$ at the end; that series will always diverge and will always approach $+\infty$, but as we have seen we can rigorously assign the value of $-1/12$ to it for the purposes of removing infinities from our calculations using, in this case, the Riemann zeta function.During the conception of this post I did ponder a question which continues to interest me, though. We saw from the example of Grandi's series that there are some mathematical operations and manipulations which would be fine in 'normal' mathematics but which suddenly become verboten when done in the specific context of a divergent series. The question is, Is this a fundamental property of the mathematics in question, that is to say, the underlying patterns and structures, or is it one emergent from notational limits? Is a rearrangment of terms in a divergent series actually fundamentally different to a rearrangment of terms in a convergent series, or is it the same thing that manifests different results in different contexts? I am not sure if this question can be answered sensibly, but for my money I am reminded of the old dichotomy in the philosophy of mathematics whose question is still yet to find an answer: Is mathematics discovered or invented? Now there is truly some food for thought.
Notes
$1$. There are other ways to use sigma notation. For example, equation \ref{eq:natural} can also be represented as $\sum_{n\leq 1}n$ ($n$ being a whole number is implied by the discrete sum being used instead of the continuous integtral) or $\sum_{n\in\mathbb{N}}n$. Other less common examples of sigma notation are $\sum_{p\text{ prime}}\frac{1}{p}$ which is the sum of the reciprocals of all prime numbers and $\sum_{d \vert n}d^x$ which is the divisor function, where '$d \vert n$' means $d$ divides $n$ exactly.$2$. The sum shown here has the interesting property of being a series representation of the number 1, or in the given notation, $\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n=1$. For any readers who are passingly familiar with Ancient Greek philosophy, this fact can be viewed as a solution to Zeno's dichotomy paradox, albeit one which ignores some nuances which I will address at a later date when I cover supertasks, one of the many intersections of philosophy and mathematics.
$3$. Formally, there exists a limit $S$ such that for any (arbitrarily small) number $\epsilon>0$ there is a number $N$ such that for $n>N$, $|S_n-S|<\epsilon$ where $S_n$ is the $n^{\text{th}}$ partial sum. Informally, there exists a number $S$ such that for an arbitrarily large $n$ the partial sum $S_n$ will be arbitrarily close to $S$.
$4$. The factorial operator is defined as $n!=n(n-1)(n-2)...1$, or in words, $n!$ (read '$n$ factorial') is given by the multiplication of $n$ by all the whole numbers less than $n$ going down to $1$. As an example, $5!=5\cdot4\cdot3\cdot2\cdot1=120$.
$5$. I will not delve here into the depths of conditional and absolute convergence, almost convergence, and so on. Suffice it to say that there are a great many interesting infinite series that have a great many interesting properties relating to convergence behaviour other than those simple ones shown here. If you are especially interested in the topic, I recommend investigating the Riemann series theorem for a very interesting ad surprising property of conditionally convergent series.
$6$. The harmonic series is deceptively interesting and there are many, many different and varied ways of calculating the harmonic numbers. One example straight out of equation \ref{eq:harmonic} is the recurrence relation (an equation which gives one term in a sequence in terms of a previous one) $H_n=H_{n-1}+1/n$. I encourage you to investigate others and see where it leads you!
$7$. The value of a field and the value of the derivative of the field at a given point in space cannot be known to arbitrary accuracy; the better one is known the less well the other must be. Though the uncertainty principle is often raised as a weird and wonderful result of quantum mechanics, in fact it is a feature of any wave theory and is linked intimately with Fourier transformations, although a thorough demonstration of how this is so is sadly beyond the scope of this note.
$8$. The complexities of quantum field theory should by no means be underestimated; what I am presenting here is an extraordinarily simplified version that, while instructive, would not necessarily be very useful in practice.
$9$. Standing waves are waves with nodes (points of zero displacement) at the endpoints. An example would be a plucked guitar string, which is retricted from moving at the bridge and nut. This restriction ensures the only possible wavelengths are given by $\lambda_n=na/2$ for length $a$ where $n$ is a natural number. Using the wave relationship $c=\nu_n\lambda_n$ we find equivalently $\nu_n=nc/2a$ (or equation \ref{eq:omega} as $\omega=2\pi\nu$) which gives the frequency $\nu$ for the $n^{\text{th}}$ harmonic.
$10$. Or rather, one of the tools, as we could equally have chosen the Abel sum, the Borel sum, the $1/x$ series method, or a number of others. Cesàro summation is far from the only rigorous way of dealing with Grandi's series, but what is important is that it gives a value of 1/2, as do the methods I listed above—this consistency is a big hint that we have picked the right number to assign to the series.
$11$. It is worth noting that if we 'dilute' the series by adding in $+0$s we change the value of the Cesàro sum (although the summability is not affected). This illuminates yet another mathematical manipulation which would be fine for a convergent series but is not for a divergent series.
$12$. Precisely because it would become divergent.
$13$. If $e$ is my favourite number then $\zeta(s)$ is surely my favourite function, but in exactly the same way I could not possibly hope to explain why except in another post devoted to it exclusively.
$14$. For fear of overwhelming you with yet more beautiful mathematics in an already over-long post I will avoid the temptation of discussing the Bernoulli numbers, although as ever I encourage the interested reader to investigate for themselves!
Thursday, February 28, 2013
Patterns, Pt. 1: Permutations and Combinations
Let me tell you a story. Once upon a
time there was a little girl who happened to like to walk. She would wander
from one end of town to the other just about every other day of the week. On
occasion her friends would ask her where she had been, and she might casually
reply, "Oh, just down to the pier," or "Oh, just down to the end
of Elm Avenue". Some of the girls would wonder in admiration how she had
the stamina for such great journeys. Others would just roll their eyes and
assume she was making the whole thing up to brag and had more likely spent the
day watching TV. The fact of the matter was, however, that there was nothing
about putting one foot in front of the other that especially resonated with our
heroine; it was just that she liked the fresh air and had come to the
realisation that she found few greater pleasures than just letting her mind
wander in peace. To each their own.
One day the girl was walking by one of her favourite ponds. Floating on the water she spied a number of leaves. Some were round, some were pointed, some were yellow and some were green. She stopped for a moment to watch them float aimlessly about on the water, and she pondered how many ways there were for a floating leaf to be. She smiled - it was a simple question, but she didn't visit ponds to challenge herself. Two different colours, two different shapes. A round leaf could be green or yellow, a pointed leaf could be green or yellow; two ways to be one way and two ways to be another makes four ways. Of course, there's nothing special about leaves, and in fact they're a little mundane. If there were three shapes there would be two ways to be one shape, two for another and two for the third - 6 ways. If there were three colours and three shapes then 9 ways. It was just simple multiplication.
The girl left the pond. She didn't much care for the floppy, soggy leaves which floated in the pond. Much better the crunchy leaves which fell on the footpath; they were so much more satisfying to step on. Soon enough she was imagining herself playing with leaves - stepping on them, kicking them around, watching the wind blow them about, crushing them with her hands, shuffling them like cards, and then finally taking a stack home in her pocket. In her mind she could see her deck of leaves sitting in her pocket, and she wondered how many ways you could put those leaves in her pocket. If she had three leaves then she could put any of the three in first, but there would only be two leaves left to choose from to put in next, and once the first two were in the final leaf was already decided by process of elimination. It struck her that this was a little like the leaves in the water - 3 'ways' for the first leaf to go in the pocket, 2 for the second and 1 for the third makes for 3×2×1=6 ways in total for her leaves to be in her pocket. Again, multiplication. How dull.
But an inquisitive mind is never satisfied so easily, and with many steps left on her trek she decided to play a game and figure out how many ways there would be for 4 leaves to go in her pocket (4×3×2×1 = 24), then 5 (5×4×3×2×1 = 120). What the little girl didn't know was that such a sequence had a name - factorial. She had never had much reason to share her thoughts with anyone else, so the idea that her sequence had a name, or should need one, had never really occurred to her. After all, she was just playing with numbers. Sadly, we do not have her luxury, so we will use the term factorial from now on and leave the playground of her mind very briefly to introduce some dreaded notation. The factorial symbol in modern mathematical notation is an exclamation mark. If one wanted to express, say, 6×5×4×3×2×1 then one might instead write 6! (six factorial). Much neater, but back to the story.
At this point the girl thought that she was sick of leaves. How many ways would there be for, say, a deck of cards to be in her pocket? 52×51×50×49×... that was a calculation far too tedious to keep her attention. She decided it was a very big number, and that satisfied her. But then she imagined two decks, then three, and then a million (because why not?). 52 million cards would probably not fit inside her pocket, even though it was quite reasonably sized. They probably wouldn't fit even if she used her other pocket too. She wondered how many cards actually would fit inside her pocket, but when she tried to think of all the ways you could stuff cards into her pocket if they weren't neatly lined up in packets it made her mind boggle. It might be a long walk, but she wasn't so bored that she would turn to the dimensions of her pockets to amuse herself. Instead she thought about packs of cards. She could probably fit four packs in a pocket. Instead of a million, suppose she had six packs. Her pocket allowed four 'slots' for her six objects to fill. 6×5×4×3. ...That was it. 360 ways. It was just another multiplication sequence like before, but truncated.
Truncating a factorial like that seemed at first a little odd to the girl. She had become enamoured with her discovery of factorials and didn't really like the idea of cutting them short. She ruminated about this cruel fact, that her perfect factorials should be ruined when the objects outnumbered the slots, and she became a little grumpy about it, perhaps more grumpy than one would ordinarily become about such things. Just as she was about to throw her hands up in the air and think about happier topics, she had a realisation. In her mind's eye she saw a line of slots and her factorial sequence filling them, 6 5 4 3, and hanging off the end like a shadow were the remnants, 2 1. But 2×1 was a factorial sequence too! The truncated sequence 6 5 4 3 was just 6 factorial with 2 factorial lopped off the end, and since each number in the sequence is multiplied, the lopping off is really just a division. Suddenly the artificial-seeming truncation could be expressed in factorials too and the girl's day was brightened up again.
Urged on by this new discovery, the girl tried to think about how to make this object-slot-fitting working for any amount of objects and any amount of slots, just in case the need should ever arise (but really because she was properly enjoying herself now). The girl didn't know much about algebra except that it was something older kids on TV seemed to complain about a lot. She did, however, understand perfectly well the notion of an unspecified amount. Because we have the advantage of modern mathematical notation, we will use n and k to represent respectively what she thought of as 'some number' and 'some other number (which might be the same as the other one but might not be)' and follow her reasoning with those replacements made. We will let n represent the number of objects and let k represent the number of slots. If that's the case then we'll start off with n! and truncate it by dividing by the leftover factorial. If there are k slots, and we are implicitly assuming that k is the same as or smaller than n, then the leftover factorial must start at n-k, resulting in the final expression n!/(n-k)!.
The girl was cautious and picked an example to make sure her expression was right.
6×5×4×3 = 360.
n = 6 (6 objects) and 6! = 6×5×4×3×2×1 = 720.
k = 4 (4 slots) and n-k = 6-4 = 2 and 2! = 2.
6!/2! = 720/2 = 360. It worked! She was over the Moon. Having long since left the pond behind she skipped along the footpath, which now took her by the local sports oval not too far from her house. She picked different numbers for n and k and checked that the answer she got was right. Every time it was, and every time she smiled gleefully. What a fun little rule she had worked out for herself!
She soon realised that there was more to the story. If she had four slots and six objects, let’s call them A, B, C, D, E and F for convenience, then even though she had a choice of 6 for the first slot, 5 for the second and so on, she could just as easily end up with A B C D one time and B A D C another, and those would be counted as two separate cases. That’s all well and good if these objects are special and it matters what order they are in, but what about if she doesn’t care what the order is? Her mind turned back to the crunchy leaves in her pocket. What is the difference between one leaf going in first and another going in second compared to the other leaf going in first and the first leaf going in second? They’re all just leaves...
And so it was apparent that if the order of the objects didn’t matter then the girl’s formula wasn’t quite right – it was over-counting ways of putting those objects into slots and she wasn’t quite sure how to change it. She turned back to her six objects A through F and her four slots. She picked four of her objects at random (not that it made a difference) and came up with A, C, D and E. If the order of the objects doesn’t matter then D E A C is just as good as A C D E or C A E D or any other ordering. But how many orderings are there? This problem seemed familiar to the girl... there were 4 objects that could go in the first slot, 3 that could go in the second... it was just another factorial! If there were k slots and k objects available then there were k! different sortings that were being counted by her formula. By her reasoning, all of the sortings for a given combination of k objects should only count once. Her old formula was n!/(n-k)!, and for any k objects there were k! different orderings, so in order for the formula to count only one ordering for those k objects rather than all of them she would have to divide by k!. That would make a new formula, n!/(k!(n-k)!). Excellent!
At this point the girl realised she was quite tired and the Sun was about to set soon, and so she headed for home, ready and eager to enjoy her dinner. This is the end of the little girl’s story, but it is not quite the end of ours. Before we retire to our own dinners we must first address this small addendum. We will find it useful in future tales to name the two expressions which the girl derived, n!/(n-k)! and n!/(k!(n-k)!), and we will name them using function notation, keeping in mind that in these formulas k is never larger than n. In actual fact there are many different notations for writing these expressions but we will not confuse ourselves by trying to use all of them.
If we have a function of two variables f(x,y) then what we mean to say is we have a machine called f which, when you input any two numbers x and y, outputs a number. For example, consider the function f(x,y) = (x+y)2. If we choose our inputs to be x = 1 and y = 2 then the machine f will spit out (1+2)2 = 32 = 9. Now, of the girl's two expressions the first we will call P(n,k), where we will use n and k for the inputs instead of x and y because we have written the cogs and gears of this machine as n!/(n-k)! which happens to involve the letters n and k (although we could just as easily have picked any other letters). The second expression we will call C(n,k). As a matter of interest, the names P and C were chosen to stand for permutation and combination. The first expression is for counting the different orderings you can make for k objects chosen from a set of n objects and to ‘permute’ two objects is to change their order. The second expression is for counting the different combinations of k objects you can make when they are chosen from a set of n objects. C in particular is going to appear again soon and not necessarily in the places you would expect, but isn’t that always the way.
One day the girl was walking by one of her favourite ponds. Floating on the water she spied a number of leaves. Some were round, some were pointed, some were yellow and some were green. She stopped for a moment to watch them float aimlessly about on the water, and she pondered how many ways there were for a floating leaf to be. She smiled - it was a simple question, but she didn't visit ponds to challenge herself. Two different colours, two different shapes. A round leaf could be green or yellow, a pointed leaf could be green or yellow; two ways to be one way and two ways to be another makes four ways. Of course, there's nothing special about leaves, and in fact they're a little mundane. If there were three shapes there would be two ways to be one shape, two for another and two for the third - 6 ways. If there were three colours and three shapes then 9 ways. It was just simple multiplication.
The girl left the pond. She didn't much care for the floppy, soggy leaves which floated in the pond. Much better the crunchy leaves which fell on the footpath; they were so much more satisfying to step on. Soon enough she was imagining herself playing with leaves - stepping on them, kicking them around, watching the wind blow them about, crushing them with her hands, shuffling them like cards, and then finally taking a stack home in her pocket. In her mind she could see her deck of leaves sitting in her pocket, and she wondered how many ways you could put those leaves in her pocket. If she had three leaves then she could put any of the three in first, but there would only be two leaves left to choose from to put in next, and once the first two were in the final leaf was already decided by process of elimination. It struck her that this was a little like the leaves in the water - 3 'ways' for the first leaf to go in the pocket, 2 for the second and 1 for the third makes for 3×2×1=6 ways in total for her leaves to be in her pocket. Again, multiplication. How dull.
But an inquisitive mind is never satisfied so easily, and with many steps left on her trek she decided to play a game and figure out how many ways there would be for 4 leaves to go in her pocket (4×3×2×1 = 24), then 5 (5×4×3×2×1 = 120). What the little girl didn't know was that such a sequence had a name - factorial. She had never had much reason to share her thoughts with anyone else, so the idea that her sequence had a name, or should need one, had never really occurred to her. After all, she was just playing with numbers. Sadly, we do not have her luxury, so we will use the term factorial from now on and leave the playground of her mind very briefly to introduce some dreaded notation. The factorial symbol in modern mathematical notation is an exclamation mark. If one wanted to express, say, 6×5×4×3×2×1 then one might instead write 6! (six factorial). Much neater, but back to the story.
At this point the girl thought that she was sick of leaves. How many ways would there be for, say, a deck of cards to be in her pocket? 52×51×50×49×... that was a calculation far too tedious to keep her attention. She decided it was a very big number, and that satisfied her. But then she imagined two decks, then three, and then a million (because why not?). 52 million cards would probably not fit inside her pocket, even though it was quite reasonably sized. They probably wouldn't fit even if she used her other pocket too. She wondered how many cards actually would fit inside her pocket, but when she tried to think of all the ways you could stuff cards into her pocket if they weren't neatly lined up in packets it made her mind boggle. It might be a long walk, but she wasn't so bored that she would turn to the dimensions of her pockets to amuse herself. Instead she thought about packs of cards. She could probably fit four packs in a pocket. Instead of a million, suppose she had six packs. Her pocket allowed four 'slots' for her six objects to fill. 6×5×4×3. ...That was it. 360 ways. It was just another multiplication sequence like before, but truncated.
Truncating a factorial like that seemed at first a little odd to the girl. She had become enamoured with her discovery of factorials and didn't really like the idea of cutting them short. She ruminated about this cruel fact, that her perfect factorials should be ruined when the objects outnumbered the slots, and she became a little grumpy about it, perhaps more grumpy than one would ordinarily become about such things. Just as she was about to throw her hands up in the air and think about happier topics, she had a realisation. In her mind's eye she saw a line of slots and her factorial sequence filling them, 6 5 4 3, and hanging off the end like a shadow were the remnants, 2 1. But 2×1 was a factorial sequence too! The truncated sequence 6 5 4 3 was just 6 factorial with 2 factorial lopped off the end, and since each number in the sequence is multiplied, the lopping off is really just a division. Suddenly the artificial-seeming truncation could be expressed in factorials too and the girl's day was brightened up again.
Urged on by this new discovery, the girl tried to think about how to make this object-slot-fitting working for any amount of objects and any amount of slots, just in case the need should ever arise (but really because she was properly enjoying herself now). The girl didn't know much about algebra except that it was something older kids on TV seemed to complain about a lot. She did, however, understand perfectly well the notion of an unspecified amount. Because we have the advantage of modern mathematical notation, we will use n and k to represent respectively what she thought of as 'some number' and 'some other number (which might be the same as the other one but might not be)' and follow her reasoning with those replacements made. We will let n represent the number of objects and let k represent the number of slots. If that's the case then we'll start off with n! and truncate it by dividing by the leftover factorial. If there are k slots, and we are implicitly assuming that k is the same as or smaller than n, then the leftover factorial must start at n-k, resulting in the final expression n!/(n-k)!.
The girl was cautious and picked an example to make sure her expression was right.
6×5×4×3 = 360.
n = 6 (6 objects) and 6! = 6×5×4×3×2×1 = 720.
k = 4 (4 slots) and n-k = 6-4 = 2 and 2! = 2.
6!/2! = 720/2 = 360. It worked! She was over the Moon. Having long since left the pond behind she skipped along the footpath, which now took her by the local sports oval not too far from her house. She picked different numbers for n and k and checked that the answer she got was right. Every time it was, and every time she smiled gleefully. What a fun little rule she had worked out for herself!
She soon realised that there was more to the story. If she had four slots and six objects, let’s call them A, B, C, D, E and F for convenience, then even though she had a choice of 6 for the first slot, 5 for the second and so on, she could just as easily end up with A B C D one time and B A D C another, and those would be counted as two separate cases. That’s all well and good if these objects are special and it matters what order they are in, but what about if she doesn’t care what the order is? Her mind turned back to the crunchy leaves in her pocket. What is the difference between one leaf going in first and another going in second compared to the other leaf going in first and the first leaf going in second? They’re all just leaves...
And so it was apparent that if the order of the objects didn’t matter then the girl’s formula wasn’t quite right – it was over-counting ways of putting those objects into slots and she wasn’t quite sure how to change it. She turned back to her six objects A through F and her four slots. She picked four of her objects at random (not that it made a difference) and came up with A, C, D and E. If the order of the objects doesn’t matter then D E A C is just as good as A C D E or C A E D or any other ordering. But how many orderings are there? This problem seemed familiar to the girl... there were 4 objects that could go in the first slot, 3 that could go in the second... it was just another factorial! If there were k slots and k objects available then there were k! different sortings that were being counted by her formula. By her reasoning, all of the sortings for a given combination of k objects should only count once. Her old formula was n!/(n-k)!, and for any k objects there were k! different orderings, so in order for the formula to count only one ordering for those k objects rather than all of them she would have to divide by k!. That would make a new formula, n!/(k!(n-k)!). Excellent!
At this point the girl realised she was quite tired and the Sun was about to set soon, and so she headed for home, ready and eager to enjoy her dinner. This is the end of the little girl’s story, but it is not quite the end of ours. Before we retire to our own dinners we must first address this small addendum. We will find it useful in future tales to name the two expressions which the girl derived, n!/(n-k)! and n!/(k!(n-k)!), and we will name them using function notation, keeping in mind that in these formulas k is never larger than n. In actual fact there are many different notations for writing these expressions but we will not confuse ourselves by trying to use all of them.
If we have a function of two variables f(x,y) then what we mean to say is we have a machine called f which, when you input any two numbers x and y, outputs a number. For example, consider the function f(x,y) = (x+y)2. If we choose our inputs to be x = 1 and y = 2 then the machine f will spit out (1+2)2 = 32 = 9. Now, of the girl's two expressions the first we will call P(n,k), where we will use n and k for the inputs instead of x and y because we have written the cogs and gears of this machine as n!/(n-k)! which happens to involve the letters n and k (although we could just as easily have picked any other letters). The second expression we will call C(n,k). As a matter of interest, the names P and C were chosen to stand for permutation and combination. The first expression is for counting the different orderings you can make for k objects chosen from a set of n objects and to ‘permute’ two objects is to change their order. The second expression is for counting the different combinations of k objects you can make when they are chosen from a set of n objects. C in particular is going to appear again soon and not necessarily in the places you would expect, but isn’t that always the way.
Tuesday, January 29, 2013
Nothingness
The question is often posed (and likely has been since questions were invented), "Why is there something rather than nothing?". This question has special philosophical and religious importance and of course I am not about to suggest that I am in any position to provide a definitive answer. With that said, I must admit that I find the question a peculiar one. In order to explain why, a slight diversion is required.
I have typically reserved this blog for musings of a strictly philosophical nature in the past, especially with regards to ethics. However, physics is my trade, and my original intention was never that the blog was to be exclusively philosophical. It is therefore with no guilt (but plenty of fore-warning) that I am about to introduce an element of physics to this particular musing, and I will have to ask that the less scientifically-minded of my readers bear with me as I will try to keep things as simple as possible.
In the popular literature one of the more commonly referred to principles of quantum mechanics is the Heisenberg Uncertainty Principle, which is commonly expressed as meaning that a particle's position and momentum cannot be simultaneously known to arbitrary precision or, as it is usually misleadingly rendered: one can either know a particle's position but not its velocity, or its velocity but not its position. Mathematically this is understood by the equation ΔxΔp ≥ ħ/2: the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) is greater than or equal to half the reduced Planck constant (a very small but importantly non-zero number).
What is less well-known is that the Uncertainty Principle applies to many other pairs of properties, perhaps the next most common pair of which is energy and time -- ΔEΔt ≥ ħ/2. This equation has far-reaching implications. Suppose we consider a small patch of empty space and contemplate the energy content there, if any. We know from the Uncertainty Principle that for any finite amount of time that you care to measure you will find your portion of 'empty' space has a decidedly non-zero energy (this energy is called the vacuum energy). This is not a question of the accuracy of your tools, it is a direct consequence of the wave-like nature of particles in quantum mechanics and is fundamentally built-in to the way the Universe works.
So our little region of space has a little bit of a buzz to it, so what? We now turn to the most famous equation in the world, Einstein's equation of mass-energy equivalence E = mc2. This suggests that from the vacuum energy matter can be created, although only in the form of virtual particles (vacuum fluctuations) with short lifespans as the Uncertainty Principle prohibits 'borrowing' energy from the vacuum indefinitely. (Sidenote: this is slightly misleading, as the relation E = mc2 only applies to real particles, with virtual particles lying slightly "off-shell" where E ≠ mc2. Even so, mass-energy equivalence remains valid and interestingly, the further off-shell a virtual particle is the shorter its lifetime, exactly as the Uncertainty Principle demands.) This means that out there in space, a sea of virtual particles are constantly flashing in and out of existence. This may sound like the ramblings of a madman, but so far as modern physics can tell it is an accepted fact.
Now, to give relevance to this slight digression, let's return to the question that sparked this blog post - why is there something rather than nothing? Well, what is nothing? A typical immediate response is to think of some small region of space deep in the void with nothing remotely nearby -- a classical vacuum. As I hope I have just demonstrated, however, this notion of nothingness is flawed as this region of space is not truly empty but is populated by virtual particles, fluctuations in the various quantum fields that permeate the universe. It seems we will have to reconsider what nothingness is, then.
Let us put aside quantum concerns for the moment. Even still the argument can be made that this does not constitute a reasonable view of nothingness. Wheeler's famous summary of general relativity (a purely classical theory) is: "Spacetime tells matter how to move; matter tells spacetime how to curve". We have forgotten that our region of empty space in a classical universe still contains, or rather consists of spacetime. In fact, we can do away with relativity altogether and still argue that we are envisioning a Universe endowed with space and time. Some may be reluctant to call these 'something', but since they have properies they are most definitely not nothing. Thus we must abandon our picture of a Universe with any notion of distance or duration, as nothingness precludes any such concept.
Now we are confronted with trying to conceive of a Universe which is literally nothing in the senses I have described above. There is so light, no matter, no particles of any description, not even space or time. Though it may seem unlikely, modern physics has the tools to deal with such a scenario, albeit in a somewhat imperfect way given our current knowledge of physics is not complete. What quantum field theory (perhaps the most accurate theory of physics ever discovered) suggests is that this nothingness is an unstable state and will inevitably and spontaneously turn into something. It seems that the way physics works just precludes nothingness from being a viable possibility. Problem solved, right?
Not quite. There is is one last resort for the hold-out, and that is to claim that nothingness is not only the absence of objects, fields, space, time and the rest, but also of physical law. While there is nothing wrong with this definition in principle, in practice it is nonsense. It only seems reasonable in my opinion that if someone is to establish their own definition of what constitutes 'nothingness' they should do so in an explicable way. This is simply not possible as it is so far outside of human experience that there is no reasonable way to explain it. Any analogy will fall short, any visualisation will falter; we had enough trouble trying to imagine a Universe with no notion of space or time!
What's more, how can we prove or disprove that such a nothingness can actually exist if we have deprived ourselves of all our tools for answering such a question? The typical recourse is to the supernatural, perhaps explaining that God somehow creates something from this absolute nothingness, which is established by hypothesis. But if this nothingness is so perfect, how can there be a God there at all? What meaning does the concept of God hold in such a scenario? Clearly He must have no physical properties as there are no physical laws, and cannot be within the Universe as it is by hypothesis nothing, so He must reside outside the nothingness, and yet be able to change it. It is at this point that I throw my hands up in the air and ask in exasperation what such a statement means, not in a deep or mystical sense but rather as a basic statement in the English language. It seems that this desperate endeavour has led us down a path which is, as I suggested earlier, nonsensical. We might as well discuss the colour of a red green, or the weight of an idea for all the progress such an approach can yield.
It may seem disingenuous to argue that we limit our definition of nothingness to what we, as flawed people, can comprehend, but in fact we have no other recourse than abandoning the question altogether. Given the curious nature of human beings, I doubt we will be able to withstand the temptation to revisit it, and so our hand is forced. It seems to me at least that we must restrict ourselves to discussing the question of why there is something rather than nothing in strictly physical terms, and that is why the question of why there is something rather than nothing strikes me as being especially peculiar in contrast to other important philosophical questions.
I have typically reserved this blog for musings of a strictly philosophical nature in the past, especially with regards to ethics. However, physics is my trade, and my original intention was never that the blog was to be exclusively philosophical. It is therefore with no guilt (but plenty of fore-warning) that I am about to introduce an element of physics to this particular musing, and I will have to ask that the less scientifically-minded of my readers bear with me as I will try to keep things as simple as possible.
In the popular literature one of the more commonly referred to principles of quantum mechanics is the Heisenberg Uncertainty Principle, which is commonly expressed as meaning that a particle's position and momentum cannot be simultaneously known to arbitrary precision or, as it is usually misleadingly rendered: one can either know a particle's position but not its velocity, or its velocity but not its position. Mathematically this is understood by the equation ΔxΔp ≥ ħ/2: the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) is greater than or equal to half the reduced Planck constant (a very small but importantly non-zero number).
What is less well-known is that the Uncertainty Principle applies to many other pairs of properties, perhaps the next most common pair of which is energy and time -- ΔEΔt ≥ ħ/2. This equation has far-reaching implications. Suppose we consider a small patch of empty space and contemplate the energy content there, if any. We know from the Uncertainty Principle that for any finite amount of time that you care to measure you will find your portion of 'empty' space has a decidedly non-zero energy (this energy is called the vacuum energy). This is not a question of the accuracy of your tools, it is a direct consequence of the wave-like nature of particles in quantum mechanics and is fundamentally built-in to the way the Universe works.
So our little region of space has a little bit of a buzz to it, so what? We now turn to the most famous equation in the world, Einstein's equation of mass-energy equivalence E = mc2. This suggests that from the vacuum energy matter can be created, although only in the form of virtual particles (vacuum fluctuations) with short lifespans as the Uncertainty Principle prohibits 'borrowing' energy from the vacuum indefinitely. (Sidenote: this is slightly misleading, as the relation E = mc2 only applies to real particles, with virtual particles lying slightly "off-shell" where E ≠ mc2. Even so, mass-energy equivalence remains valid and interestingly, the further off-shell a virtual particle is the shorter its lifetime, exactly as the Uncertainty Principle demands.) This means that out there in space, a sea of virtual particles are constantly flashing in and out of existence. This may sound like the ramblings of a madman, but so far as modern physics can tell it is an accepted fact.
Now, to give relevance to this slight digression, let's return to the question that sparked this blog post - why is there something rather than nothing? Well, what is nothing? A typical immediate response is to think of some small region of space deep in the void with nothing remotely nearby -- a classical vacuum. As I hope I have just demonstrated, however, this notion of nothingness is flawed as this region of space is not truly empty but is populated by virtual particles, fluctuations in the various quantum fields that permeate the universe. It seems we will have to reconsider what nothingness is, then.
Let us put aside quantum concerns for the moment. Even still the argument can be made that this does not constitute a reasonable view of nothingness. Wheeler's famous summary of general relativity (a purely classical theory) is: "Spacetime tells matter how to move; matter tells spacetime how to curve". We have forgotten that our region of empty space in a classical universe still contains, or rather consists of spacetime. In fact, we can do away with relativity altogether and still argue that we are envisioning a Universe endowed with space and time. Some may be reluctant to call these 'something', but since they have properies they are most definitely not nothing. Thus we must abandon our picture of a Universe with any notion of distance or duration, as nothingness precludes any such concept.
Now we are confronted with trying to conceive of a Universe which is literally nothing in the senses I have described above. There is so light, no matter, no particles of any description, not even space or time. Though it may seem unlikely, modern physics has the tools to deal with such a scenario, albeit in a somewhat imperfect way given our current knowledge of physics is not complete. What quantum field theory (perhaps the most accurate theory of physics ever discovered) suggests is that this nothingness is an unstable state and will inevitably and spontaneously turn into something. It seems that the way physics works just precludes nothingness from being a viable possibility. Problem solved, right?
Not quite. There is is one last resort for the hold-out, and that is to claim that nothingness is not only the absence of objects, fields, space, time and the rest, but also of physical law. While there is nothing wrong with this definition in principle, in practice it is nonsense. It only seems reasonable in my opinion that if someone is to establish their own definition of what constitutes 'nothingness' they should do so in an explicable way. This is simply not possible as it is so far outside of human experience that there is no reasonable way to explain it. Any analogy will fall short, any visualisation will falter; we had enough trouble trying to imagine a Universe with no notion of space or time!
What's more, how can we prove or disprove that such a nothingness can actually exist if we have deprived ourselves of all our tools for answering such a question? The typical recourse is to the supernatural, perhaps explaining that God somehow creates something from this absolute nothingness, which is established by hypothesis. But if this nothingness is so perfect, how can there be a God there at all? What meaning does the concept of God hold in such a scenario? Clearly He must have no physical properties as there are no physical laws, and cannot be within the Universe as it is by hypothesis nothing, so He must reside outside the nothingness, and yet be able to change it. It is at this point that I throw my hands up in the air and ask in exasperation what such a statement means, not in a deep or mystical sense but rather as a basic statement in the English language. It seems that this desperate endeavour has led us down a path which is, as I suggested earlier, nonsensical. We might as well discuss the colour of a red green, or the weight of an idea for all the progress such an approach can yield.
It may seem disingenuous to argue that we limit our definition of nothingness to what we, as flawed people, can comprehend, but in fact we have no other recourse than abandoning the question altogether. Given the curious nature of human beings, I doubt we will be able to withstand the temptation to revisit it, and so our hand is forced. It seems to me at least that we must restrict ourselves to discussing the question of why there is something rather than nothing in strictly physical terms, and that is why the question of why there is something rather than nothing strikes me as being especially peculiar in contrast to other important philosophical questions.
Sunday, May 8, 2011
Update... Sort of
Hey guys.
You've no doubt noticed that I haven't updated in a while.
I was trying to prevent that from happening by instituting weekly updates.
I chose Wednesdays and Thursdays because I ordinarily update least on those days.
Well... turns out there was a reason for that. So I haven't been updating. At all.
And the fact of the matter is that I'm busy getting busier, so I'm not going to be updating for a while.
I can't say this won't be a permanent thing, but there are still plenty of things I want to say.
Time will tell.
See you all soon!
You've no doubt noticed that I haven't updated in a while.
I was trying to prevent that from happening by instituting weekly updates.
I chose Wednesdays and Thursdays because I ordinarily update least on those days.
Well... turns out there was a reason for that. So I haven't been updating. At all.
And the fact of the matter is that I'm busy getting busier, so I'm not going to be updating for a while.
I can't say this won't be a permanent thing, but there are still plenty of things I want to say.
Time will tell.
See you all soon!
Saturday, March 19, 2011
The parable of the vinegar tasters
Another short tale, this time based on a traditional Chinese painting subject, also known as the Vinegar Tasters.
Three wise men were travelling across the country, going from town to town to teach wisdom. At one town in particular, they were welcomed warmly by the townspeople, and offered gifts of the town's produce. One of the gifts was a large pot of vinegar. The first of the wise men dipped his finger into the vinegar, tasted it, and a sour expression came across his face. "This vinegar is too sour," he said, "take it away." The second of the wise men dipped his finger into the vinegar, tasted it, and a scowl came across his face. "This vinegar tastes terrible. It is far too bitter, take it away." Then the third man dipped his finger into the vinegar, tasted it, and smiled. "Mmm!" The other two looked at him confused. "Now this tastes like vinegar!"
In the Chinese tradition, the three vinegar tasters are Confucius, Buddha and Loazi, and each one's reaction is representative of their respective religion's attitude on life. Confucius sees the vinegar as soured wine, and sees it in need of correction, just as society should be maintained through strict rules. Buddha recoils at the taste of the vinegar, and sees it as being too extreme for the body and an example of the suffering of life. Laozi, on the other hand, recognises that vinegar is what it is, and our perception of it is biased by our judgement of what it "should be".
In the same way, all indifferents, be they goods such as vinegar or money or fine arts or even medicine, or events such as celebrations or tragedies or deaths, are all as they are and nothing more, neither inherently good nor bad. Recognise them as such and you will see how your perception has been coloured, and know where you have mistakenly placed your values. Only then will you be able to accept yourself as part of the harmony of Nature, rather than willing yourself against it.
Three wise men were travelling across the country, going from town to town to teach wisdom. At one town in particular, they were welcomed warmly by the townspeople, and offered gifts of the town's produce. One of the gifts was a large pot of vinegar. The first of the wise men dipped his finger into the vinegar, tasted it, and a sour expression came across his face. "This vinegar is too sour," he said, "take it away." The second of the wise men dipped his finger into the vinegar, tasted it, and a scowl came across his face. "This vinegar tastes terrible. It is far too bitter, take it away." Then the third man dipped his finger into the vinegar, tasted it, and smiled. "Mmm!" The other two looked at him confused. "Now this tastes like vinegar!"
In the Chinese tradition, the three vinegar tasters are Confucius, Buddha and Loazi, and each one's reaction is representative of their respective religion's attitude on life. Confucius sees the vinegar as soured wine, and sees it in need of correction, just as society should be maintained through strict rules. Buddha recoils at the taste of the vinegar, and sees it as being too extreme for the body and an example of the suffering of life. Laozi, on the other hand, recognises that vinegar is what it is, and our perception of it is biased by our judgement of what it "should be".
In the same way, all indifferents, be they goods such as vinegar or money or fine arts or even medicine, or events such as celebrations or tragedies or deaths, are all as they are and nothing more, neither inherently good nor bad. Recognise them as such and you will see how your perception has been coloured, and know where you have mistakenly placed your values. Only then will you be able to accept yourself as part of the harmony of Nature, rather than willing yourself against it.
Friday, March 18, 2011
Thursday's Thought for the Week: 17 Mar 2011
I've read a few translations of Chapter 23, from which this week's WQW was taken. The original text of the Tao Te Ching was written in Ancient Chinese, which is notoriously difficult to translate, and so you can find for many of the poems inside a very wide variety of translations, some with very different and even contradictory meanings. Of the translations I read of Chapter 23, they varied from the very metaphorical to the very literal. I decided to go with one of the more literal versions as a jumping off point for this week's TTW, only because it better serves me in talking about the particular subject of this post, that being the topic of communication, although I could as well have used it to speak about a number of other topics had I decided to go with a differing interpretation.
The quote I provided yesterday seems fairly explicit. Nature is capable of making great statements (if you will) not only despite their being brief, but also because they are brief. Humans too, the reasoning goes, should endeavour to be so forceful and succinct with their words. Few people, especially in this day and age of constant interaction, would argue that point much. Email, instant messaging, Facebook, all those confounded doo-dads the older generations dismiss are a double-edged swords at times, especially now that we carry them all around in our pockets. Aloneness is almost a foreign concept, at least where such technologies are prevalent.
Consider also, if you please, another very old saying along a similar vein. It suggests that there is an inversely proportional relationship to how much you talk and how much you ought to talk. I would give you some direct quotes, but history is so replete with examples and variations on the theme, that it would give me great astonishment that someone reading this blog should never had come across it before. The point is, I imagine anyone who has come across a statement to this effect would not lodge any disagreement with it, at least not in principle, unless perhaps that person happened to be the argumentative type. They may even nod sagely in view of this great wisdom, maybe with a knowing smile in someone or other's direction, and lament the way our society has turned towards this need for constant stimulation.
This is a sad state indeed, largely because such people are heedless of this advice with astonishing flair. Now, I'm not about to whinge about people being too undisciplined to actually follow through with advice they know is worthwhile, because goodness, it's not as though I haven't made my opinion on that fact before. No, I would much rather take the simpler recourse and ask people to actually go out of their way and try to, every now and then, shut up.
Those who know me closely will likely scoff at the idea of me being a particularly quiet person, as I can be as unnecessarily boisterous as anyone, given the right conditions, but with that said when I'm with people I'm not very close with, even if they're friends, I'm much less likely to be as loud or talkative. A lot of people tell me that this is some sort of a problem, as though my reluctance to say something in such a scenario is some sort of extreme shyness or antisociability, and I should try and be more "outgoing". The fact of the matter is, however, that if I am with people I don't know, or don't know well, then if I don't have much of substance to say I'm not terribly inclined to talk. There's very little point in flapping your lips around and making noise that nobody particularly cares to listen to, don't you think?
Now, if there's something to be said, or if someone asks you a question, there's little to be accomplished in staying quiet, or being abrupt. That is just as unproductive as talking when you shouldn't. Know when to speak and when to stay silent, and I think you will find that you learn much more from listening to other people than you could from taking over conversations or perpetuating inane and pointless chatter. Live and let live, as they say - if there is business to conduct, then conduct it. If not, relax, enjoy yourself, but remember that like many other things, the more words you offer, the cheaper each word becomes.
On a related topic, and incidentally the initial thought behind this post, is the matter of essay-writing and speech-giving. If you're ever in a position to be writing essays or giving speeches, you will likely be trying to persuade someone, and it is a matter of common knowledge that a good way to persuade someone is to impress them. Following this line of thought, people can and often are tempted to pull out a thesaurus and start using the longest synonyms they can find at every opportunity, even though they may not be aware of the common usage such words may hold, or any particular connotations. This is often a bad idea because people often see such people as either being genuinely unpleasant intellectuals or insincerely unpleasant poseurs attempting to look smarter than they are, but that is not why I bring it up - this is not a blog about essay-writing techniques.
What point I would much rather push is the idea that you shouldn't go out of your way to use big words for the simple reason that it gets in the way of your communication. Now, obviously, if you're referring to a very specific concept you want to get across with some accuracy, then by all means be as technical as need be, but if you're trying to say a very mundane thing in a very round-about way, then all you are doing is wasting time, wasting breath and wasting those precious words I was talking about before. Not only that, but all people should value the truth (I hold that to be a self-evident truth for the time being). If you are using words that your audience doesn't understand, then you are obscuring your argument; you not only make it harder to understand your sentences, you make it harder to understand your arguments, your points, and to do that is to obscure the truth. The truth is, in many cases, as plain as it is simple.
Am I saying you should never lie? Maybe yes, maybe no. That is not the point of this post. Am I saying that we should always be completely upfront and honest with people, no matter how they feel? Possibly, but again, that is not the point of this post. The point of this post is not what people say, but how they choose to say it, and how free they feel to do so. Save your words for things that need to be said. Be forceful. Let yourself have meaning in your speech, not only when you address a crowd or an authority, but at all times. Do not devalue your ideas by throwing them into a sea of nonsense and triviality that serves no purpose other than to alleviate some ill-founded notion of awkwardness, or social mores.
Or perhaps I'm just rambling.
The quote I provided yesterday seems fairly explicit. Nature is capable of making great statements (if you will) not only despite their being brief, but also because they are brief. Humans too, the reasoning goes, should endeavour to be so forceful and succinct with their words. Few people, especially in this day and age of constant interaction, would argue that point much. Email, instant messaging, Facebook, all those confounded doo-dads the older generations dismiss are a double-edged swords at times, especially now that we carry them all around in our pockets. Aloneness is almost a foreign concept, at least where such technologies are prevalent.
Consider also, if you please, another very old saying along a similar vein. It suggests that there is an inversely proportional relationship to how much you talk and how much you ought to talk. I would give you some direct quotes, but history is so replete with examples and variations on the theme, that it would give me great astonishment that someone reading this blog should never had come across it before. The point is, I imagine anyone who has come across a statement to this effect would not lodge any disagreement with it, at least not in principle, unless perhaps that person happened to be the argumentative type. They may even nod sagely in view of this great wisdom, maybe with a knowing smile in someone or other's direction, and lament the way our society has turned towards this need for constant stimulation.
This is a sad state indeed, largely because such people are heedless of this advice with astonishing flair. Now, I'm not about to whinge about people being too undisciplined to actually follow through with advice they know is worthwhile, because goodness, it's not as though I haven't made my opinion on that fact before. No, I would much rather take the simpler recourse and ask people to actually go out of their way and try to, every now and then, shut up.
Those who know me closely will likely scoff at the idea of me being a particularly quiet person, as I can be as unnecessarily boisterous as anyone, given the right conditions, but with that said when I'm with people I'm not very close with, even if they're friends, I'm much less likely to be as loud or talkative. A lot of people tell me that this is some sort of a problem, as though my reluctance to say something in such a scenario is some sort of extreme shyness or antisociability, and I should try and be more "outgoing". The fact of the matter is, however, that if I am with people I don't know, or don't know well, then if I don't have much of substance to say I'm not terribly inclined to talk. There's very little point in flapping your lips around and making noise that nobody particularly cares to listen to, don't you think?
Now, if there's something to be said, or if someone asks you a question, there's little to be accomplished in staying quiet, or being abrupt. That is just as unproductive as talking when you shouldn't. Know when to speak and when to stay silent, and I think you will find that you learn much more from listening to other people than you could from taking over conversations or perpetuating inane and pointless chatter. Live and let live, as they say - if there is business to conduct, then conduct it. If not, relax, enjoy yourself, but remember that like many other things, the more words you offer, the cheaper each word becomes.
On a related topic, and incidentally the initial thought behind this post, is the matter of essay-writing and speech-giving. If you're ever in a position to be writing essays or giving speeches, you will likely be trying to persuade someone, and it is a matter of common knowledge that a good way to persuade someone is to impress them. Following this line of thought, people can and often are tempted to pull out a thesaurus and start using the longest synonyms they can find at every opportunity, even though they may not be aware of the common usage such words may hold, or any particular connotations. This is often a bad idea because people often see such people as either being genuinely unpleasant intellectuals or insincerely unpleasant poseurs attempting to look smarter than they are, but that is not why I bring it up - this is not a blog about essay-writing techniques.
What point I would much rather push is the idea that you shouldn't go out of your way to use big words for the simple reason that it gets in the way of your communication. Now, obviously, if you're referring to a very specific concept you want to get across with some accuracy, then by all means be as technical as need be, but if you're trying to say a very mundane thing in a very round-about way, then all you are doing is wasting time, wasting breath and wasting those precious words I was talking about before. Not only that, but all people should value the truth (I hold that to be a self-evident truth for the time being). If you are using words that your audience doesn't understand, then you are obscuring your argument; you not only make it harder to understand your sentences, you make it harder to understand your arguments, your points, and to do that is to obscure the truth. The truth is, in many cases, as plain as it is simple.
Am I saying you should never lie? Maybe yes, maybe no. That is not the point of this post. Am I saying that we should always be completely upfront and honest with people, no matter how they feel? Possibly, but again, that is not the point of this post. The point of this post is not what people say, but how they choose to say it, and how free they feel to do so. Save your words for things that need to be said. Be forceful. Let yourself have meaning in your speech, not only when you address a crowd or an authority, but at all times. Do not devalue your ideas by throwing them into a sea of nonsense and triviality that serves no purpose other than to alleviate some ill-founded notion of awkwardness, or social mores.
Or perhaps I'm just rambling.
Wednesday, March 16, 2011
Wednesday's Quote of the Week: 16 Mar 2011
I know I said that I'd try and do this every week, and I know I failed at that after one week, but in my defence I've been extremely busy the past two weeks and I don't normally think of blogging on Wednesdays and Thursdays anyway. Of course, that's why I chose those days, but knowing me, it may have been a poor decision in hindsight. Irrespective, here is your quote of the week for this week:
This quote is taken from Chapter 23 of the Ursula K. Le Guin translation of the Tao Te Ching by Lao Tzu [Dao De Jing by Laozi]. A follow-up post should come tomorrow.
Nature doesn't make long speeches.
A whirlwind doesn't last all morning.
A cloudburst doesn't last all day.
Who makes the wind and rain?
Heaven and earth do.
If heaven and earth don't go on and on,
certainly people don't need to.
This quote is taken from Chapter 23 of the Ursula K. Le Guin translation of the Tao Te Ching by Lao Tzu [Dao De Jing by Laozi]. A follow-up post should come tomorrow.
Saturday, March 5, 2011
A note on another old adage
Following on from my earlier musings on old sayings, another one which came to mind today was one I'm sure you're familiar with: every cloud has a silver lining. For those of you who haven't come across it, it basically means that every bad situation in your life has some positive aspect about it. Like the earlier example, this too is said far too often with far too little thought. In fact, it strikes me that there are a very many sayings supposed to make miserable people chipper that more often than not only serve to annoy them, but I digress.
I don't think I need to go into too much depth analysing the saying. The premise is fairly simple, and seems reasonable enough, that nothing is ever all bad (and I agree with that). Some people seem to take it to an odd extreme, however, and wander through their lives with inane grins, seeing nothing but argentian skies. They seem especially fond of telling others that every cloud has a silver lining, no matter what should have befallen them. Then there are those who trudge their way through life grim and depressed, sure that the sky is blocked out by heavy rain clouds, and any silver lining is merely lightning in the distance. I don't think I need to say that these people are equally as right as the other lot, in the sense that they are not at all.
If you ask my opinion, and since you're reading this I assume you do, then there aren't really any clouds and nor are there any silver linings. If things are looking bleak, it's probably because you're looking down at your own shadow. If things are looking overwhelmingly peachy, you may be staring up at the sun. As always, the key lies in the mean; keep your eyes level, focussed, forward and un-wandering, and you will see what is.
I don't think I need to go into too much depth analysing the saying. The premise is fairly simple, and seems reasonable enough, that nothing is ever all bad (and I agree with that). Some people seem to take it to an odd extreme, however, and wander through their lives with inane grins, seeing nothing but argentian skies. They seem especially fond of telling others that every cloud has a silver lining, no matter what should have befallen them. Then there are those who trudge their way through life grim and depressed, sure that the sky is blocked out by heavy rain clouds, and any silver lining is merely lightning in the distance. I don't think I need to say that these people are equally as right as the other lot, in the sense that they are not at all.
If you ask my opinion, and since you're reading this I assume you do, then there aren't really any clouds and nor are there any silver linings. If things are looking bleak, it's probably because you're looking down at your own shadow. If things are looking overwhelmingly peachy, you may be staring up at the sun. As always, the key lies in the mean; keep your eyes level, focussed, forward and un-wandering, and you will see what is.
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