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I'm pretty new to QFT. So is it possible, before delving into the history and motivation and variations and implications and perturbation techniques and so on, to briefly state the postulates in just one coherent way? That is, to assert every mathematical object that is in play, what it represents, and the mathematical relations between them?

For example, QED is the simplest QFT, right? And I want to avoid operators for now if possible. So here's my guess, from the bits I've read over the years:

  1. Spacetime: There is a Minkowski space $M$, representing spacetime.
  2. Electron Field: At every point $x$ of $M$, there is a Dirac spinor $\psi$ (4 complex numbers, that transform a certain way under Lorentz transforms). This represents the probability density that any spin-up/spin-down electron/positron is at $x$.
  3. Photon (EM) field: At every point of $M$, there is also a 4-vector $A$ -- of real numbers? -- that transforms like a 4-vector. This represents the electromagnetic potential.

Are those all the objects? And then the law, relating the objects:

  1. There is a certain function of the electron and photon fields, $\psi$ and $A$, called the Lagrangian. The 'action' $S_p$ along a path $p$ of $M$ is the path integral of this function.
  2. Pick two complete, disjoint, spacelike hypersurfaces $T$ and $U$ of $M$. First of all, the integral of the electron field on $T$ and $U$ must match, and this represents the number of electrons & positrons in the universe/experiment.
  3. Then, the value of the electron field at each point $y$ of $U$ is proportional to the integral, a la Feynman, over all imaginable paths $p$ between every point $x$ of $T$, and $y$, of $\psi(x)\exp(\frac{i}{\hbar}S_p)$. This, together with (5), determines the evolution of the electron field.

Is that close? Maybe not, because among other things it doesn't say how the photon field evolves. But isn't it possible to state the theory in one such consistent way? Is there a source, preferably online, that lays it out?

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    $\begingroup$ I assume you're talking about relativistic QFT. Even in that case, QED is not the simplest QFT. One of the simplest is the QFT of a single scalar field, such as the free scalar field (exactly solvable) or the $\phi^4$ model (not exactly solvable AFAIK). These models are easier than QED by far. Other essential warm-ups for QED are the free Dirac spinor field (without electromagnetism) and the free Maxwell field (without spinors). All of these can be formulated using a path-integral. Would you rather start with one of those? Or do you want the answer to be about QED? $\endgroup$ Apr 25 '21 at 1:39
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    $\begingroup$ @ChiralAnomaly Well that's certainly informative, and undoubtedly I'll end up having to go back and look at those simpler models. But at the same time, since I understand Dirac spinors, electromagnetism, and Feynman's path integral concept, I can't help feeling like there should be a straight statement of QED in those terms, even if it's not as elegant as other ways of stating it. But if I'm way off base, by all means point me to the simpler ones. $\endgroup$ Apr 25 '21 at 1:51
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    $\begingroup$ Starting with QED and then jumping back as needed is a perfectly reasonable approach. Yes, there is a straight statement of QED in path-integral terms. One warning: it's not strictly well-defined in continuous spacetime, but it's good enough to convey the idea. If you want it to be strictly well-defined, then the only known ways of doing that (nonperturbatively) involve discretizing spacetime, but that's messy and artificial, so I'd recommend tucking that away in the back of your mind for now and starting with the continuous-spacetime version. $\endgroup$ Apr 25 '21 at 2:06
  • $\begingroup$ @ChiralAnomaly Haha, it's always something crazy like that, isn't it? Okay, I'm game! $\endgroup$ Apr 25 '21 at 2:15
  • $\begingroup$ Are Osterwalder-Schraeder axioms what you're looking for? $\endgroup$ Apr 25 '21 at 3:15
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I'll sketch the postulates for one version of a path-integral formulation of QED. My goal here is only to give a brief orientation, with emphasis on a few conceptual points that many introductions neglect. For more detail, here are a few on-line resources:

I'm sure there are lots of others, too. I didn't find any that mention the specific path-integral formulation described below, which gives the time-evolution of an arbitrary state, but there are lots of resources about the path integral that generates all time ordered vacuum expectation values, which uses mostly the same ideas.

Ingredients

The ingredients are much like what the question listed. I'll list them again with a little extra clarification.

  • Spacetime: Morally, we're thinking of continuous four-dimensional Minkowski spacetime $M$. But for the sake of clarity, we can consider a very fine discretization of spacetime, with a very large but finite extent, so we can think of $M$ as a finite set, say with a mere $10^{100000}$ points. For doing numerical calculations, we would need to trim that down a bit, but that's not the goal here.

  • Matter field: At every point $x\in M$, there is a set of four Grassmann variables (not complex numbers) denoted $\psi_k(x)$ and four more Grassmann variables denoted $\overline\psi_k(x)$. In constrast to real or complex numbers, Grassmann variables all anticommute with each other. They don't have values, but we can still define functions (polynomials) of them, and we can still do something analogous to integrating over them (called the Berezin integral), which we use to define the path integral. This is important: they don't represent probability densities. Probability densities are functions of the field variables. More about this below.

  • EM field: For every pair of neighboring points in $M$, say $x$ and its nearest neighbor in the $\mu$-direction, we associate a single complex variable $e^{i\epsilon A_\mu(x)}$, where $A_\mu(x)$ is a real variable (one for each $x$ and each $\mu$) and $\epsilon$ is the step-size between neighboring points. When we write the continuum version of the action, we use only $A_\mu(x)$ instead of $e^{i\epsilon A_\mu(x)}$, but some concepts are easier in the discretized version, so I wanted to mention it here.

Those are (adjusted versions of) the ingredients listed in the question. To help make contact with the general principles of quantum theory, which still hold in quantum field theory, I'll add two more:

  • Observables: Observables represent things that could be measured. In QFT, observables are expressed in terms of the field operators — field variables and derivatives with respect to the field variables. In QED, observables are required to be invariant under gauge transformations, at least under those that are continuously connected to the identity (trivial) gauge transformation. One example of an observable is $F_{ab}\equiv \partial_a A_b-\partial_b A_a$, and another example is $\overline\psi_j(y)e^{i\int_x^y ds\cdot A(s)}\psi_k(x)$. In the second example, the $A$-dependent factor can be regarded as a product of the complex variables $e^{i\epsilon A_\mu(x)}$ along some path from $x$ to $y$.

  • States: To specify a state, we specify a function $\Psi$ of all of the field variables associated with some spacelike slice, say the $t=0$ slice. I'll write this as $\Psi[\psi,\overline\psi,A]_0$, with a subscript to indicate which time the field variables are restricted to. I'm using square brackets as a standard reminder that $\Psi$ is a function of an enormous number of variables — several per point in the given spacelike slice. States, like observables, are required to be gauge-invariant.

A common point of confusion among QFT newcomers is the relationship between the state and the field variables, especially in QED where the spinor field is traditionally denoted $\psi$ — the same symbol that is also traditionally used for the state. These are not the same thing. I'm using an uppercase $\Psi$ for the state and a lowercase $\psi$ for the spinor field. The distinction should become clear below, if it isn't already.

Time evolution

Most of the literature about the path-integral formulation of QED focuses on time-ordered vacuum expectation values of products of field variables. That's useful for many reasons, but here I'll use a path-integral formulation to describe time-evolution in the Schrödinger picture starting from a given initial state (which is not necessarily the vacuum state), because this seems to be closer to what the question requested.

The path integral formulation describes how the state evolves in time. Schematically, the law looks like this: $$ \Psi_\text{final}[\psi,\overline\psi,A]_t = \int [d\psi]_{(t,0]}[d\overline\psi]_{(t,0]}[dA]_{(t,0]}\ \exp\Big(iS[\psi,\overline\psi,A]_{[t,0]}\Big) \Psi_\text{initial}[\psi,\overline\psi,A]_0 $$ where $S[\psi,\overline\psi,A]$ is the action — the integral of the Lagrangian over the part of spacetime between times $0$ and $t$. Defining the integrals mathematically takes more work. Here, I'm only highlighting one detail: the subscripts $(t,0]$ and $[t,0]$ indicate what part of spacetime the integration variables come from. In words: The initial state depends on the field variables in the time$=0$ slice. The action depends on all of the field variables from time$=0$ through time$=t$, inclusive. The integral is over all of these except the ones at the final time $t$. The result is a new state that depends only the variables associated with the time=$t$ slice.

Probabilities

Quantum field theory is just a special kind of quantum theory. The general principles of quantum theory, namely Born's rule and the projection rule, still apply. However, unlike nonrelativistic single-particle models in which the state is a function of the spatial coordinates, here the state is a function of an enormous number of field variables. Each such function, if it's sufficiently well-behaved, represents a state-vector in the Hilbert space.

Observables are operators on the Hilbert space, which can be expressed using multiplication by field variables and derivatives with respect to field variables. This is analogous to how operators are usually expressed in nonrelativistic single-particle models, but with a huge number of field variables in place of three spatial coordinates. In quantum field theory, the variables are field variables, and the spacetime coordinates play the role of indices that we use to keep track of all those field variables.

Schematically, given some observable $X$, its expectation value in a normalized state $\Psi$ is $$ \langle \Psi|X|\Psi\rangle \sim \int [d\psi][d\overline\psi][dA]\ \Psi^*[\psi,\overline\psi,A] X \Psi[\psi,\overline\psi,A]. $$ Again, the observable $X$ is some combination of multiplication by field variables and derivatives with respect to field variables. If $X$ is a projection operator onto one of an observable's eigenspaces, then this expectation value is the probability of getting that outcome if the observable is measured.

I'm glossing over the definitions of the integrals, but conceptually, the inner product $\langle\Psi_1|\Psi_2\rangle$ is an integral over the field variables on which the states $\Psi_1$ and $\Psi_2$ depend, just like in more familiar single-particle models — except that now the variables are field variables instead of spatial coordinates.

Wait — what about particles?

In QED, particles (electrons/positrons/photons) are phenomena that the theory predicts. The theory is expressed in terms of fields, not particles. The theory includes observables that act as particle detectors, and it has single- and multi-particle states, but constructing them explicitly is prohibitively difficult unless we resort to perturbation theory, which is exactly what most (all?) introductions do. Perturbation theory is a whole other industry, and I won't try to cover it here.

I will say one thing about particles. Field variables are indexed by (equivalently, are functions of) spacetime, so we can define their positive/negative frequency parts. If we set the coupling constant to zero, which makes the model boring, then we can calculate those parts explicitly. Whether or not we can calculate them explicitly, the positive-frequency parts act as energy-reducing operators (they annihilate the vacuum state), and the negative-frequency parts act as energy-increasing operators. In the zero-coupling case, they annihilate and create individual particles. Given any state $\Psi$, hitting it with the negative-frequency part of a field variable adds a particle — either a photon if the field is $A$ or as an electron or positron if the field is $\overline\psi$ or $\psi$, respectively. Beware that this simple relationship between fields and particles is restricted to the zero-coupling case — or to perturbation theory, which I won't go into here.

Other things I glossed over

I glossed over lots of other things, too. For example, I didn't explain how to define the integral over a Grassmann variable. I didn't say anything about how Wick rotation can be used to relate the arbitrary-initial-state formulation to the vacuum-expectation-value formulation, which dominates most textbooks for a good reason. I also didn't say anything about other axiomatic approaches to QFT, each of which brings its own valuable perspectives.

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    $\begingroup$ I really appreciate it, both the clarifying comments and the exquisitely detailed answer. Much obliged. $\endgroup$ Apr 25 '21 at 21:58

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