301. Relating field theories via stochastic quantization

Posted by Domenico Orlando

March 11, 2009

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Domenico Orlando, a full-time member of IPMU (Japan), is known for his works on many aspects of string theory such as string theory on curved backgrounds, quantum crystals and exact solutions in string theory. Dmitry.

First of all I would like to thank Dmitry for inviting me to write about Stochastic Quantization in connection with a recent paper (arXiv:0903.0732) by Robbert Dijkgraaf, Susanne Reffert and myself.

In one sentence, Stochastic Quantization is a quantization scheme, introduced by Parisi and Wu, in which quantum mechanics is seen as the thermal equilibrium of a stochastic process with respect to an extra (fictious) time dimension. But let me start from the beginning.

The main idea. The main idea behind Stochastic Quantization is that using the analogy between the Euclidean path integral measure and the Boltzmann distribution for a statistical system in equilibrium one can write the Euclidean Green’s functions as limits of equal-time correlators for an appropriate stochastic process. To see how to do this in practice, let us consider as an example a scalar field theory in d dimensions, described by a field $\phi(x)$ with action $S_{\text{cl}}[\phi]$ . To construct the associated stochastic process we do the following:

introduce a new time dimension t, so that $\phi$ is promoted to a function of (d+1) variables $\phi (t,x)$ ;
impose a time evolution for $\phi(t,x)$ such that for $t \to \infty$ the system goes to equilibrium and “describes the same physics” as the quantum d-dimensional system. A simple way of doing this consists in imposing a Langevin equation
$\partial_t \phi(t,x)=– \frac{1}{2} \frac{\delta S_{\text{cl}}}{\delta \phi} + \eta (t,x)$

where $\eta(t,x)$ is a white Gaussian noise.

One can then show that for $t \to \infty$ the averages over the noise $\eta$ tend to the path integral averages for the d-dimensional system. That is,

$\lim_{t \to \infty} \langle \phi(t,x_1) \phi(t,x_2) \dots \rangle_\eta=\langle \phi(x_1) \phi(x_2) \dots \rangle$ .

Up to this point this is by now a well known construction. What we argue in our work is that a completely analogous approach can be followed to quantize a discrete system, such as a spin chain. In this case, starting from a system that can assume a set of configurations $\alpha$ with energy $H(\alpha )$ we add the time direction t and impose a master equation for the probability to be in the configuration $\alpha$ at time t:

$\partial_t P_{\alpha }(t)=\sum_{\beta \neq \alpha } W_{\alpha \beta} P_\beta (t) – W_{\beta \alpha } P_\alpha (t)$ ,

where the transition rates are written as

$W_{\alpha \beta}=e^{- \left( H(\alpha ) – H(\beta) \right)}$

if the system can pass from $\beta$ to $\alpha$ in a single elementary step (such as flipping a spin).

One can then prove that for $t \to \infty$ , the probability $P_\alpha (t)$ tends to the unique ground state of the matrix which corresponds – just like before – to the Boltzmann weight of the initial “classical” system

$\lim_{t \to \infty} P_\alpha (t)=P_\alpha^{(0)} \propto e^{-H(\alpha)}$ .

Now you might wonder: why should I use stochastic quantization? What is the advantage of this quantization procedure? There are different types of answers to this question:

from a purely technical point of view it turns out that stochastic quantization is particularly useful in the case of systems with a lot of symmetries since it does not require gauge fixing;
this approach is naturally suited for numerical simulations of large systems;
one can decide (and this is what we do in our paper) to take the extra time dimension seriously and study the (d+1)-dimensional systems.

Lagrangian and Hamiltonian description. In order to understand the dynamics in (d+1) dimensions better it is useful to translate the Langevin (or Master Equation) into a Lagrangian (resp. Hamiltonian) formalism:

choosing a Lagrangian approach one can see that it is natural to introduce fermionic partners of the field $\phi$ , collect them into a supermultiplet $\Phi$ and write the (d+1)-dimensional action in a manifestly supersymmetric:
$S_q[\Phi]=D \Phi {\bar D} \Phi + S_{\text{cl}}[\Phi]$ ,

where D is the supercovariant derivative and $S_{cl}[\Phi]$ now has taken the role of a superpotential. This points to a deep connection between Brownian motion and supersymmetry that is unfortunately beyond the scope of this post.
Choosing the Hamiltonian approach we found that in the discrete case, the system is described by a Schrodinger equation and the Hamiltonian is precisely the Laplacian on the state graph, i.e. the graph whose nodes are the classical configurations and whose lines join two configurations if they are connected by an elementary move (see the picture). This means that for $t \to \infty$ , the probability amplitudes are described by a harmonic function on the graph:
$H_{\text{q}} | \Psi_0 \rangle=\nabla^2 | \Psi_0 \rangle=0$ .

It is also a common property of all these models that the norm squared of the ground state is precisely the classical partition function in d dimensions:

$\langle \Psi_0 | \Psi_0 \rangle=Z_{\text{cl}}$ .

State Graph

Some examples. In our paper we consider a number of examples of pairs of theories in d and (d+1) dimensions related by Stochastic Quantization. Interestingly enough some of these pairs where already known but not recognized as being connected by this quantization scheme. More precisely we describe in some detail

zero-dimensional field theory and one-dimensional super quantum mechanics;
d-dimensional free bosons and (d+1)-dimensional super quantum Lifshitz model;
classical and quantum crystals (and dimers);
the XXZ model as growth of integer partitions;
gauged WZW model and Chern Simons (with some caveats).

I would like to end this post concentrating on one of these examples. Our initial motivation for this paper (as string theorists) comes from the observation made some time ago (see hep-th/0309208) that the partition function for the topological string A-model on $\mathbf{C}^3$ is the same as the partition function for the classical three-dimensional crystal melting. This is a stochastic system in which cubes are stacked in an empty corner of 3D space and a cube can be added to a configuration if three of its sides will touch either the wall or other cubes (see figure).

On the one hand, the Stochastic Quantization of this system is very interesting in itself: in particular in one dimension less (where one considers squares instead of cubes), we proved in a previous work (arXiv:0803.1927) that the quantized system is precisely the integrable XXZ spin chain with kink boundary conditions.
On the other hand we argue that the construction is also relevant in string theory since, according to the correspondence with topological strings, each cube configuration corresponds to a geometry. Then the quantum wavefunctions are naturally interpreted as sums over geometries and the Laplace equation for the ground state can be read in the spirit of the Wheeler-de Witt equation.

To summarize: Stochastic Quantization is a quantization scheme that relates a system in d dimensions to a stochastic process in (d+1). In our paper we argue that it can be also used for discrete systems and we describe some examples of theories in different dimensions that are related by this scheme. These examples are of interest for different fields, ranging from solid state physics to string theory.

Further Reading

The original paper (quite difficult to find): G. Parisi and Y.-S. Wu, Perturbation Theory Without Gauge Fixing, Sci. Sin. 24 (1981) 483
A collection of relevant papers: P.H. Damgaard and H. Huffel: “Stochastic quantization’‘ (on google book search)
A comprehensive review: P. H. Damgaard and H. Huffel, Stochastic quantization, Phys. Rep. 152 (1987), no. 5-6 227-398.
A more recent book: M. Namiki: “Stochastic quantization”

Journal club, Master Mason, Quantum field theory, String theory

If you liked the post, please kindly consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you want to chat with me in real time, you can find me on Twitter. The posts below are probably related to the subject of this one:

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4 Comments »

Comment by Lubos Motl

2009-03-11 23:55:26

That’s a pleasing paper, including the other two authors whom I know, unlike D.O. Kimigayo to Susanne, whatever it means.

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Comment by Max

2009-03-12 06:02:36

Hi!
Interesting paper. One more illustration of duality Markovian-Hamiltonian. The consequence of linearity.

White Gaussian noise. Why?

Thanks.
Best regards.

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Comment by Domenico Orlando

2009-03-12 06:43:11

Thanks.

The choice of a WGN is essentially to make your life easier. As long as the stochastic process converges to equilibrium you are free to choose the source.
As a matter of fact, in some cases such as fermionic theories, you are actually forced to introduce different distributions (kernels).

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