Quantum particles and classical filaments

[From_Branes_to_Brains]

The first post in the “From Branes to Brains” series. For the introduction to the series see this post.

In this post, I will mostly follow the ideas from the classical book by Polyakov, as well as from this paper by Kleinert.

First of all, consider a classical particle with mass $m$ in the one-dimensional potential $V(x,t)$. If we know, that at the moment $t_1$ the particle was in the point $x_1$, and in the moment $t_2$ the particle was in the point $x_2$, the particle should have travelled between these two points along the path that minimizes the action

$S\left[x\right(t),x_1,x_2,t_1,t_2]={\int }_{t_1}^{t_2}dt(\frac{m}{2}{\left(\frac{dx\left(t\right)}{dt}\right)}^2-V\left(x\right(t),t))$

This is known as the minimal action principle. In quantum mechanics, instead, the particle travels along all the possible trajectories, according to the Feynman principle. Namely, the amplitude of the probability of the transition from point $x_1$ at the moment $t_1$ to the point $x_2$ at the moment $t_2$ is given by:

$⟨x_1,t_1|x_2,t_2⟩={\int }_{x\left(t_1\right)=x_1,x\left(t_2\right)=x_2}Dx\left(t\right)\exp \left(\frac{i}{\hslash }S\right[x\left(t\right),x_1,x_2,t_1,t_2\left]\right)$

The Feynman-Hibbs book is quite a good introduction to this topic. Now let us consider a system, that seems to be totally different: classical inextensible flexible filament (like a guitar string or long polymer molecule) in the two-dimensional potential $V(x,y)$ under tension $F$. The string is pinned in points $x_1,y_1$ and $x_2,y_2$, and there are length reservoirs at this point. To visualize, you can think about string on the table that passes through the holes in the table in points $x_1,y_1$ and $x_2,y_2$, and there is tension $F$ applied to the ends. The energy of such string is total potential energy plus $\tau L$, where $L$ is the length of the string on the table.
If the angle of the string tangential with $Y$ axis is everywhere small, the length is approximately given by the formula

$L={\int }_{y_1}^{y_2}dy[1+\frac{1}{2}{\left(\frac{dx\left(y\right)}{dy}\right)}^2]$

where $x\left(y\right)$ controls the shape of the filament.
The total energy of the string is then

$E\left[x\right(y),x_1,x_2,y_1,y_2]={\int }_{y_1}^{y_2}dy[\frac{F}{2}{\left(\frac{dx\left(y\right)}{dy}\right)}^2+V(x\left(y\right),y\left)\right]$

where we ignored the term 1 since it just gives a constant contribution to the energy, without changing anything.
And the partition function is then

$Z={\int }_{x\left(y_1\right)=x_1,x\left(y_2\right)=x_2}Dx\left(y\right)\exp [-\frac{1}{T}E[x\left(y\right),x_1,x_2,y_1,y_2\left]\right]$

where each term in the sum corresponds to the probability of a given configuration at a temperature $T$. One may already see some similarities between the expression for a classical string and a quantum particle. We can, however, make it even more transparent.

Let’s assume that the quantities we calculate have all the good mathematical properties necessary for their safe continuation to the complex plane. Then we can look at the amplitude for the quantum particle transition, assuming time to be imaginary $t=-i\tau$ where $\tau$ is real. This may look like black magic, but it is a quite useful approach in quantum field theory. It is called Wick rotation, or rotation to Euclid (because this change turns time into the usual space coordinate). Then the transition amplitude turns to

$⟨x_1,{\tau }_1|x_2,{\tau }_2⟩={\int }_{x\left({\tau }_1\right)=x_1,x\left({\tau }_2\right)=x_2}Dx\exp [-\frac{1}{\hslash }{\int }_{{\tau }_1}^{{\tau }_2}d\tau (\frac{m}{2}{\left(\frac{dx}{d\tau }\right)}^2+V(x,\tau ))]$

Now we see the exact correspondence. The amplitude for the quantum particle transition in the imaginary time is equal to the partition function for the classical string at the temperature $1/\hslash$ with length $t_2-t_1$ and tension $m$.

It is not clear, whether there is a fundamental reason for this fact, or it is just a coincidence. For our purposes, however, it is not important. What we see is that we can borrow from quantum mechanics its apparatus, and use it to describe the directed polymer under tension—quite a common ingredient in the biopolymer filament networks, that forms the carcass of living cells (cytoskeleton), and tissues together with the cells.

Of course, this is not a whole story. First of all, many filaments are not absolutely flexible. They are semiflexible, i.e., they prefer to be straight—when they are bent, there is an additional contribution to the energy. This can be written as

$E\left[x\right(y),x_1,x_2,y_1,y_2]={\int }_{y_1}^{y_2}dy[\frac{\kappa }{2}{\left(\frac{d^{2}x\left(y\right)}{d{y}^2}\right)}^2+F\frac{1}{2}{\left(\frac{dx\left(y\right)}{dy}\right)}^2+V(x\left(y\right),y\left)\right]$

where the first term corresponds to the bending energy. This is totally novel thing in comparison with quantum mechanics—there are no higher-order derivatives there, and there are fundamental reasons for that. However, it does not mean that we can not study such problems using the same mathematical apparatus, and, for example, Kleinert shows how to do it.

Second, in most cases, our filament is not confined to two dimensions, but in three dimensions. This is easily treatable by considering corresponding quantum particles in 2+1 dimensions (i.e., two spatial plus time, while before we had one spatial and one time).

Third, here we assumed that filament does not deviate from the straight state a lot. In the real world, many polymers do significantly deviate from their straight state. This would correspond to relativistic particles.

Fourth, here we were talking about just a zero-dimensional object (quantum particle) turning into a one-dimensional object (classical filament). What about more complicated stuff? For example, would string theory map onto membranes? Or, generally speaking, quantum $D$-dimensional object on classical—$D+1$dimensional object at a particular temperature.

In some sense, yes. However, real-world membranes are significantly more complex than quantum strings. Understand me correctly, I am not saying that string theory is simple in comparison with the modern theory of membranes. The only thing that I am saying is that quantum string would be a membrane without any bending, and totally homogeneous. Thus, while one can borrow some apparatus from strings, as well as from two-dimensional gravity and (for the case of three-dimensional elastic solids) from three-dimensional gravity, this is not the whole story. Certainly, this is a good start, but the addition of bending and anisotropy adds much more spices to the problem.

To sum up, we saw how problems about the behavior of elastic objects at a certain temperature, so common in biology, maps onto the quantum mechanical problems.

What I plan for the future posts (I can change plans if I get interesting suggestions).

-how Ising model from condensed matter helps study disease spread and forest fires

-how unsupervised learning using deep neural network exactly maps to renormalization procedure, the 50-year-old technique from both statistical and high energy physics.

-not only quantum: Casimir and Schwinger effects are effects from quantum field theory, but they have analogs where thermal fluctuations play the role of quantum fluctuations.

-gene surfing: physics of stochastical processes decribes the gene spread in the population

-suggest a topic yourself!

[Charlie Steiner]

Fun so far!

I know about Wick rotation, but I’m curious about the converse—have you ever heard of someone using imaginary temperature?

It’s not even all that crazy—the Boltzmann distribution would just be sinuosidal. A cursory googling also turns up the notion of expanding phase diagrams of simple lattice models to the complex temperature plane. But I’ve never heard of an actual case where this would show up (unlike negative temperature which has justifying examples like lasers).

[From_Branes_to_Brains]

Thank you!
As a technique, there is so-called Matsubara frequency calculation using something like imaginary temperature (not exactly, but it is the closest thing I can think about).

From the physical point of view, the imaginary temperature would mean complex probabilities, which is a very strange thing. From a frequentist approach, they do not make sense at all. From bayesian approach, we potentially can define it, but still not clear what would they mean.

[Charlie Steiner]

I figure it would be the same as complex electric fields etc—you just take the real part at the end.

[From_Branes_to_Brains]

Ok, but what would negative probability then be?

[Charlie Steiner]

Good question :P