What is the program of quantum field theory? What is its derivation?

Question

Paraphrasing Griffith's: For some particle of mass m constrained to the x-axis subject to some force $F(x,t)=-∂V/∂x$, the program of classical mechanics is to determine the particle's position at any given time: $x(t)$. This is obtained via Newton's second law $F=ma$. $V(x)$ together with an initial condition determines $x(t)$.

The program of quantum mechanics is to obtain the particle's wave function $\Psi(x,t)$, gotten from solving the Schrôdinger equation:

$$i \hbar \frac{∂\Psi}{∂t} = -\frac{\hbar^2}{2m}\frac{∂^2\Psi}{∂x^2} + V\Psi .$$

This is a simple case, but it illustrates the program, and generalizes to multiple particles, 3 dimensions, spin, and magnetism easily.

What is the equivalent program of quantum field theory?

And also, what is the specific representation of the "state" within that program? For example, in quantum mechanics for 1 particle in 3 dimensions, excluding spin, $\Psi: R\times R^3 \rightarrow C $ subject to normalization constraints.

Another property of the previous two programs is that it is immediately clear how the state variables evolve numerically over time (if not calculable).

And for such a solution program, is there an algebraic derivation, the way the Galilean group provides such a derivation for the Schrôdinger equation in quantum mechanics?

I'm aware of second quantization, and that particle number changes, and I've seen various Langrangians, but only for specific cases, and these are unsatisfying compared to the seemingly generic programs of other branches.

An answer dependent on Hamiltonian mechanics, classical field theory, exterior calculus, or abstract algebra is fine.

Edit: This is not a duplicate. I've seen the other question, and it's getting at how QFT differs from single-particle QM generally. I'm asking what is the specific solution program that is just generic enough to encompass all of quantum field theory, and incidentally the mathematical structure of the instances of the state variables in it, and also incidentally whether an algebraic derivation of the program exists.

Answer 1

"The Program of QFT" might be specified as follows. Specify a Lagrangian $L$ for the particle content of the model in question. For instance, a scalar field of mass $m$ would be $$L=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi+\frac{1}{2}m\phi^2$$ or a Fermion field might be $$L=i\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi-m\bar{\Psi}\Psi$$ (those are four-component spinors, but it seems like you don't care about that so much). The generating function for the scalar field looks like $$Z(J)=\int\mathcal{D}\phi e^{i\int dx^4[L+J\phi]},$$ and now we can ask questions like "what's the amplitude for the field to travel from $x_1$ to $x_2$?". The answer is $$\langle0|T\phi(x_2)\phi(x_1)|0\rangle=\frac{1}{i}\frac{\delta}{\delta J(x_2)}\frac{1}{i}\frac{\delta}{\delta J(x_1)}Z[J] \Big|_{J=0}$$ (taking most of my notation from Shrednicki). You can add more fields to your Lagrangian and ask more complicated questions of your generating procedure, but this is the heart of the program (ignoring so, so, so many details, renormalization being a major one).

If you'd like an algebraic formulation, you can think of the canonical commutation relations. There is a Hamiltonian associated to that Lagrangian I wrote above, and each field has a canonically conjugate momentum (let's use Weyl spinors this time), $$\pi^a(x)=\frac{\partial L}{\partial \dot{\psi}_a(x)}$$ which obeys (anti)commutation relations $$\{\psi_a(x),\psi_b(y)\}=0,\quad \{\psi_a(x),\pi^b(y)\}=i\delta^b_a \delta(x-y)$$

The equation of motion in this case is the Dirac one. In general cases you must also implement constraints on these commutation relations, following the Dirac paradigm (check out LQG for that stuff, they are great at it).

This might be sweeping most of the meat under the rug, but I think looking at what you've called "the program of quantum mechanics" above, I think this is a reasonably short description of the program of QFT.

Answer 2

Programs of physics can be posed in a unified mathematical formulation as follows.

Consider a parameter space M and a configuration space C. The program of physics is to calculate histories x: M->C

1. For non-relativistic mechanics we take M to be a time interval so M=R. For a single particle C=R3 its position in space.

2. For relativistic mechanics we switch to using a proper time interval M=R. For a single particle C=RxR3, its position in Lorentzian space-time.

3. For relativistic field theory we switch again to make space-time the parameter space M=RxR3 and C=R say for a real field theory. Now we are calculating field histories x: RxR3 -> R.

The classical program of the above three cases is to solve a problem of variational calculus to find a unique history x:M->C for a suitable action functional S(x) (given suitable initial conditions).

The quantum program of the above three cases is to calculate a functional integral over a space of histories x:M->C (with each history weighted by exp(iS(x)/h)).

The quantum program reduces to the classical program as h->0.

To explore the mathematical challenges involved following this style I refer to Towards the Mathematics of Quantum Field Theory, Frederic Paugam (2014).

Answer 3

I think it may be helpful for the answer to take a look at an intermediate theory. Namely, consider a scalar field, and, instead of this field being defined on $\mathbb{R}$ or $\mathbb{R}^3$, using a lattice on $\mathbb{Z}$ or $\mathbb{Z}^3$, or, even better, a finite lattice on on $\mathbb{Z}_N$ or $\mathbb{Z}_N^3$ with periodic boundary conditions. In the last case, you are simply in the same situation as in standard quantum mechanics, except that the dimension is not 3 but $N$ resp. $N^3$.

If everything is fine with $\mathbb{Z}_N^3$ lattice theory, then it remains to care about renormalization. The lattice theory is a valid regularization of the continuous field theory. So, for a larger enough N, the lattice theory is a good enough approximation of QFT anyway. And if the limit makes sense at all is an open question. Look for Haag's theorem, which, very roughly, claims that the limit makes no sense.

Whatever, for $\mathbb{Z}_N^3$ lattice theory the answer is the same as for standard quantum theory.