The path integral has many applications:
Mathematical Finance:
In mathematical finance one is faced with the problem of finding the price for an "option."
An option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or sell a specified asset, the underlying, on or before a specified future date, the option's expiration date, at a given price, the strike price. For example, an option may give the buyer the right but not the obligation to buy a stock at some future date at a price set when the contract is settled.
One method of finding the price of such an option involves path integrals. The price of the underlying asset varies with time between when the contract is settled and the expiration date. The set of all possible paths of the underlying in this time interval is the space over which the path integral is evaluated. The integral over all such paths is taken to determine the average pay off the seller will make to the buyer for the settled strike
price. This average price is then discounted, adjusted for for interest, to arrive at the current value of the option.
Statistical Mechanics:
In statistical mechanics the path integral is used in more-or-less the same manner as it is used in quantum field theory. The main difference being a factor of $i$.
One has a given physical system at a given temperature $T$ with an internal energy $U(\phi)$ dependent upon the configuration $\phi$ of the system. The probability that the system is in a given configuration $\phi$ is proportional to
$e^{-U(\phi)/k_B T}$,
where $k_B$ is a constant called the Boltzmann constant. The path integral is then used to determine the average value of any quantity $A(\phi)$ of physical interest
$\left< A \right> := Z^{-1} \int D \phi A(\phi) e^{-U(\phi)/k_B T}$,
where the integral is taken over all configurations and $Z$, the partition function, is used to properly normalize the answer.
Physically Correct Rendering:
Rendering is a process of generating an image from a model through execution of a computer program.
The model contains various lights and surfaces. The properties of a given surface are described by a material. A material describes how light interacts with the surface. The surface may be mirrored, matte, diffuse or any other number of things. To determine the color of a given pixel in the produced image one must trace all possible paths form the lights of the model to the surface point in question. The path integral is used to implement this process through various techniques such as path tracing, photon mapping, and Metropolis light transport.
Topological Quantum Field Theory:
In topological quantum field theory the path integral is used in the exact same manner as it is used in quantum field theory.
Basically, anywhere one uses Monte Carlo methods one is using the path integral.
Best Answer
This question is quite general, I'll write just my own point of view, and hope others add more to get a complete picture.
0) Let me quote A. Kirillov himself:
Sorry I cannot find another much more colorful quote from him, where he says something like orbit method not only produced results on many principal questions of representation theory, but gives informal guidance how to invent new and new results.
1) What is the context of orbit method and why it is related to mathematical physics. Orbit method as a particular case of quantization ideology.
I think orbit method should be seen in the context of quantization and roughly speaking its relation to the mathematical physics is that orbit method is particular example of "quantization program". (Well, there some other relations with integrable systems, but they are not so central, imho).
Let me try to explain this in more details. Consider universal enveloping $U(g)$, it is an algebra with generators $e_k$ and relations $[e_{k}, e_{l} ] = c^{i}_{kl} e_l $, let us insert parameter $h$ here: $[e_{k}, e_{l} ] = h c^{i}_{kl} e_l$. For $h=0$ we have just the commutative algebra - denote it $S(g)$, for any non-zero $h$ this algebra is isomorphic to $U(g)$.
Now let us look at $h$ "very small", i.e. we can formally take $h^2=0$, what will see from the structure of non-commutative algebra $U(g)$ reduces to commutative algebra $S(g)$ "plus" a Poisson bracket on it.
So the moral is that Non-commutative algebra, when non-commutativity tends to zero is commutative algebra with Poisson bracket. (It is called classical limit).
The big goal of quantization program is try to express everything about non-commutative algebra in terms of
Poisson algebra. The reason is that Poisson algebra is something more simple than non-commutative algebra.
In particular we can be interested in description of irreps of non-commutative algebra. What are the corresponding objects for the Poisson algebra ? Answer - symplectic leaves. Observation: the symplectic leaves of $S(g)$ (classical limit of $U(g)$) are exactly the coadjoint orbits.
So this puts the orbit method - in more general framework of quantization. Where one may hope to describe the irreps of quantized algebras via symplectic leaves in Poisson algebra. The problem is that such construction does not always exists neither for $U(g)$, nor for general quantum algebras, at least it does not exist in some simple sense, and big activity is to understand what are the borders between true statements from fakes, it is neverending activity.
It is worth to remark that such point of view on orbit method is not the original one, but emerged later when quantization theory begin to develop.
The basic question of representation theory are calculating characters, induction-restriction, tensor products. The natural question: what are the parallel constructions in the Poisson world (i.e. symplectic geometry) ? how representation theory questions can be answered with the help of symplectic geometry ? There are ideological answers to these questions and again it is neverending game to make "ideology" to theorems or to counter-examples.
Some MO-questions with more details on quantization: Q1, Q2.
2) What is naive quantization without metaplectic correction. (It is better to call it correction (imho), but not quantization, but people use both).
So, our basic wish is to construct a irreducible representation of $U(g)$ (or more generally of some quantized algebra $\hat A$). The "naive recipe" is the following:
1) take symplectic leaf
2) Consider algebra of functions on it and split it into two halves "P-part" and "Q-part
3) The representation space is ----- all functions of "Q"-variable, and representation is constructed as: Q-variables act as multiplication operators, while "P" acts as $\partial_Q$.
Now, what I mean "split in two parts", informally you should think as follows Darboux theorem says that symplectic form is $dp\wedge dq$ is appropriate coordinates, so you have these "p" and "q" as my splitting. The problem is that Darboux is local result, and you need something more complicated to make it work, look at the word "polarization" for more information on that.
3) Towards metaplectic correction.
In the previous item I wrote that naively we should take "half of functions" on the orbit(symplectic leaf) as a Hilbert space.
This actually a point to be corrected.
We should not take "functions", but should take "half-forms".
The simplest motivation is that we want to have a Hilbert space, so we need to have an inner product, but there is no canonical one on the functions. But if we take half-forms: $f(q) \sqrt{dq}$ we have the canonical inner product: $\int fg \sqrt{dq}^2 = \int fg dq $.
So the metaplectic correction is story is how to introduce these "half-forms" into a business in an appropriate way. Some time ago we exercised with quantization of sphere $S^2$ and argued that in general this should be consistent with the Duflo isomorphism.
4) Finally to your main question: "quantization of coadjoint orbits".
Well, sorry, I cannot say much. The general ideology here is that we should take a coadjoint orbit and try to construct irrep. Kirillov done it in 1962 for nilpotent groups, for solvable much progress achieved later. For semisimples - generic orbits - no problems, but for many orbits it is impossible (or at least in some naive sense impossib). I do not know what is the current state of art. It might be there are some particular classes of orbits where some people think that one can indeed construct irrep and it is in reach and good topic for a paper of a PhD. It might be that ideas by Ranee Brylinski Geometric Quantization of Real Minimal Nilpotent Orbits can be somehow developed... But I do not know much about it, my impression that all left open problems are quite difficult and technical, I would not start this as a PhD. Any case I would ask David Vogan or Jeffrey Adams (he is sometimes on MO). By the way have a look at D. Vogan's REVIEW OF “LECTURES ON THE ORBIT METHOD,”.
Any way good luck !