Introduction to Perturbative Quantum Field Theory - Comments

[A. Neumaier]

By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here

• pAQFT 1: A first idea of quantum fields

I need to digest the concept of a smooth set employed in your setting. Are there relations to the Conceptual Differential Calculus of Wolfgang Bertram? (This exists in a number of variants, one of them being in https://arxiv.org/abs/1503.04623 .) Since this is somewhat off-topic here, I asked a corresponding questions at PhysicsOverflow; please reply there.

My plan had been to simply port this code here to Physics Forums. Unfortunately, this turns out to be impractical, due to numerous syntax changes that would need to be made.

With Greg we are looking for a solution now. A technically simple solution would be to simply include that webpage inside an "iframe" within the PF-Insights article. But maybe this won't be well received with the readership here? If anyone with experience in such matters has a suggestion, please let me know.

I always converted by hand, though it takes a considerable amount of time.

 

[DrDu]

Interesting article!
You say that pQFT is a perturbational expansion not only in coupling constant but also in Plancks constant. The latter point is not immediately clear to me.

 

[vanhees71]

It is since the number of loops counts the powers of $\hbar$. This is clear from the path-integral formalism since you can understand the Dyson series also as saddle-point approximation of the path integral. See, e.g., Sect. 4.6.6 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

 

[Demystifier]

Happily, no experiment occurs in an infinite laboratory, so IR divergences are a mere calculation inconvenience (it is not very practical to perform analytic calculations with big but finite IR cutoffs), not a genuine physical problem.

 

[Urs Schreiber]

You say that pQFT is a perturbational expansion not only in coupling constant but also in Planck's constant. The latter point is not immediately clear to me.

Here is how to see it:

The explicit $\hbar$-dependence of the perturbative S-matrix is

$$
S(g_{sw} L_{int} + j_{sw} A)
=
T \exp\left(
\tfrac{1}{i \hbar}
\left(
g_{sw} L_{int} + j_{sw} A
\right)
\right)
\,,
$$

where $T(-)$ denotes time-ordered products. The generating function

$$
Z_{g_{sw}L_{int}}(j_{sw} A)
\;:=\;
S(g_{sw}L_{int})^{-1} \star S(g_{sw}L_{int} + j_{sw} A)
$$

involves the star product of the free theory (the normal-ordered product of the Wick algebra). This is a formal deformation quantization of the Peierls-Poisson bracket, and therefore the commutator in this algebra is a formal power series in $\hbar$ that, however, has no constant term in $\hbar$ (but starts out with $\hbar$ times the Poisson bracket, followed by possibly higher order terms in $\hbar$):

$$
[L_{int},A] \;=\; \hbar(\cdots)
\,.
$$

Now by Bogoliubov's formula the quantum observables are the derivatives of the generating function$$
\hat A
:=
\tfrac{1}{i \hbar} \frac{d}{d \epsilon}
Z_{g_{sw}L_{int}}(\epsilon j A)\vert_{\epsilon = 0}
$$

Schematically the derivative of the generating function is of the form

$$
\begin{aligned}
\hat A
& :=
\tfrac{1}{i \hbar} \frac{d}{d \epsilon}
Z_{g_{sw}L_{int}}(\epsilon j A)\vert_{\epsilon = 0}
\\
& =
\exp\left(
\tfrac{1}{i \hbar}[g_{sw}L_{int}, -]
\right)
(j A)
\end{aligned}
\,.
$$

(The precise expression is given by the "retarded products", see (Rejzner 16, prop. 6.1).)
By the above, the exponent $\tfrac{1}{\hbar} [L_{int},-]$ here yields a formal power series in $\hbar$, and hence so does the full exponential.Here is how this relates to loop order in the Feynman perturbation series:

Each Feynman diagram $\Gamma$ is a finite labeled graph, and the order in $\hbar$ to which this graph contributes is

$$
\hbar^{ E(\Gamma) - V(\Gamma) }
$$

where

1. $V(\Gamma) \in \mathbb{N}$ is the number of vertices of the graph
2. $E(\Gamma) \in \mathbb{N}$ is the number of edges in the graph.

This comes about (see at S-matrix -- Feynman diagrams and Renormalization for details) because

1) the explicit $\hbar$-dependence of the S-matrix is

$$
S\left(\tfrac{g}{\hbar} L_{int} \right)
=
\underset{k \in \mathbb{N}}{\sum} \frac{g^k}{\hbar^k k!} T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )
$$

2) the further $\hbar$-dependence of the time-ordered product $T(\cdots)$ is

$$
T(L_{int} L_{int}) = prod \circ \exp\left( \hbar \int \omega_{F}(x,y) \frac{\delta}{\delta \phi(x)} \otimes \frac{\delta}{\delta \phi(y)} \right) ( L_{int} \otimes L_{int} )
\,,
$$

where $\omega_F$ denotes the Feynman propagator and $\phi(x)$ the (generic) field observable at point $x$ (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as scalar fields, because this is all that affects the counting of the $\hbar$ powers).

The resulting terms of the S-matrix series are thus labeled by

1. the number of factors of the interaction $L_{int}$, these are the vertices of the corresponding Feynman diagram and hence each contibute with $\hbar^{-1}$

2. the number of integrals over the Feynman propagator $\omega_F$, which correspond to the edges of the Feynman diagram, and each contribute with $\hbar^1$.

Now the formula for the Euler characteristic of planar graphs says that the number of regions in a plane that are encircled by edges, the faces, here thought of as the number of "loops", is

$$
L(\Gamma) = 1 + E(\Gamma) - V(\Gamma)
\,.
$$

Hence a planar Feynman diagram $\Gamma$ contributes with

$$
\hbar^{L(\Gamma)-1}
\,.
$$

So far this is the discussion for internal edges. An actual scattering matrix element is of the form

$$
\langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right)
\vert \psi_{in} \rangle
\,,
$$

where

$$
\vert \psi_{in}\rangle
\propto
\tfrac{1}{\sqrt{\hbar^{n_{in}}}}
\phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle
$$

is a state of $n_{in}$ free field quanta and similarly

$$
\vert \psi_{out}\rangle
\propto
\tfrac{1}{\sqrt{\hbar^{n_{out}}}}
\phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle
$$

is a state of $n_{out}$ field quanta. The normalization of these states, in view of the commutation relation $[\phi(k), \phi^\dagger(q)] \propto \hbar$, yields the given powers of $\hbar$.

This means that an actual scattering amplitude given by a Feynman diagram $\Gamma$ with $E_{ext}(\Gamma)$ external vertices scales as

$$
\hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 }
\,.
$$

 

[A. Neumaier]

1) the explicit $\hbar$-dependence of the S-matrix is

in the formula you sum over $k$ but the factors have an $n$-dependence!

 

[Urs Schreiber]

in the formula you sum over $k$ but the factors have an $n$-dependence!

Thanks! Fixed now.

 

[A. Neumaier]

Thanks! Fixed now.

You could turn post #40 with little extra work into a self-contained Insight article. It would display the information more prominently, easier to find again, and easier to reference.

 

[Urs Schreiber]

You could turn post #40 with little extra work into a self-contained Insight article.

Right, that would be part of my plans for a later chapter of the series that is meant to be started here. But I would like to proceed systematically from the beginning and first have some articles on the basics.

I'll try to slowly-but-surely work through converting to PF-Insights-formatting the material that I have already prepared. It might be going slow, though. I am still hoping some kind soul could be found who would write a simple script that could autmatically convert my source code. This should be a triviality for experts versed in such matters...

 

[A. Neumaier]

Right, that would be part of my plans for a later chapter of the series that is meant to be started here. But I would like to proceed systematically from the beginning and first have some articles on the basics.

I'll try to slowly-but-surely work through converting to PF-Insights-formatting the material that I have already prepared. It might be going slow, though. I am still hoping some kind soul could be found who would write a simple script that could autmatically convert my source code. This should be a triviality for experts versed in such matters...

Actually I find the category-theoretic language that you employ in your second part quite intimidating, whereas the content of post #40 is of the same abstraction level as part 1 (which we discuss here) and hence accessible to a wider audience. I can understand thre categorial stuff only by translating item by item to something more concrete, whereas reading stuff at the abstraction level of standard quantum field theory is much easier to read. Thus putting the category-heavy stuff later, or accompanying it by a detailed interpretation key for nonexperts in categories, might enhance understandability a lot.

 

[Urs Schreiber]

Actually I find the category-theoretic language that you employ in your second part quite intimidating,

Hm. There is intentionally zero category theory, but only basic differential geometry, in the part to follow !

I am guessing what you mean is that you clicked on the links to "smooth set" and found material not meant for your eyes. You don't need this.

The beginning of the part to follow, the section titled "Geometry" is meant to be all that you need to know for handling differential forms on spaces of field histories in a simple and yet precise manner. Maybe we could have some back and forth on it here, so that I see which words I should add.

Here is all there is to it:

a "smooth set" (or "diffeological space") is defined simply by declaring which maps from $\mathbb{R}^n$s into it are supposed to be smooth. There is some evident consistency conditions on such a declaration, but it just says what you intuitively expect to hold true anyway, so you need not bother on the first go.

For example the smooth structure on the set $[\Sigma,F]$ of smooth functions from spacetime $\Sigma$ to some field fiber $F$ is defined like so:

For $U = \mathbb{R}^n$ some Cartesian space, then a function

$$ \Phi_{(-)} : U \longrightarrow [\Sigma,F]$$

is declared to be smooth precisley if the corresponding function in both variables

$$ \Phi_{(-)}(-) : U \times \Sigma \longrightarrow Y$$

is smooth in the ordinary sense.

That's it. Compare to what you need to go through to define Frechet manifold structure or similar on this space. This here is dead easy. And it is a theorem that this is all we actually need.

The only other thing to know is now how to define a differential form on a smooth set. Like so: it's a system of ordinary differential forms on Cartesian spaces, one for each map out of the latter into our "smooth set" that we declared to be a smooth function. The only condition is that under composition of smooth functions these forms pull along compatibly. That's it.

For example if $[\Sigma,F]$ is the smooth mapping space as above, then a differential n-form $\omega \in \Omega^n([\Sigma,F])$ is for each Cartesian space $U$ and each smooth function $\Phi_{(-)} \colon U \to [\Sigma,F]$ an ordinary differential n-form on $U$, suggestively denoted
$\left(\Phi_{(-)}\right)^\ast \omega \in \Omega^n(U)$, such that for every ordinary smooth function $V \overset{f}{\to} U$ from another Cartesian space $V$, we have the evident consistency relation $(\Phi_{f(-)})^\ast \omega = f^\ast (\Phi_{(-)})^\ast \omega$ between ordinary differential forms on $V$.

That's precisely all background on differential geometry of mapping spaces that is needed for the notes . Nothing more. In particular no category theory is needed (the category theory is doing its work silently in the background, but you need not worry about that).

 

[A. Neumaier]

Here is all there is to it:

a "smooth set" (or "diffeological space") is defined simply by declaring which maps from RnRn\mathbb{R}^ns into it are supposed to be smooth. There is some evident consistency conditions on such a declaration, but it just says what you intuitively expect to hold true anyway, so you need not bother on the first go.

Yes, I looked at smooth sets since this was the first new term in the ''part to follow''. I didn't know it so I had to look it up - and the explanation given was mystifying. Your new explanation sounds simpler but it is not a mathematical definition but only handwaving. To turn it into rigorous math one needs to specify the properties demanded from a collection of smooth maps to deserve the name so that the foundation is sound. Apparently you need not just one smooth set but a whole collection (or category?) of smooth sets closed under Cartesian products and who knows what else, to be able to formulate this.

And it is a theorem that this is all we actually need.

I'd like to have a precise formulation of this theorem and what precisely is needed.

 

[Urs Schreiber]

To turn it into rigorous math one needs to specify the properties demanded from a collection of smooth maps to deserve the name so that the foundation is sound.

Sure if you want to check full details from the foundations, you need to look at the full details starting from the foundations. But to just read the notes on QFT, you can take this as a black box. But the details are also easy to check, the category theory is minimal. It's really true that the theory of smooth sets is completely elementary certianly as compared to the heavy analytic machinery required for the usual infinite-dimensional manifold structures.

Apparently you need not just one smooth set but a whole collection (or category?) of smooth sets closed under Cartesian products and who knows what else, to be able to formulate this.

Nah. You need as many smooth sets as you would otherwise consider smooth manifolds. One for spacetime, one for the field bundle, one for its jet bundle, one for the space of sections, one for the phase space. Just the obvious spaces that are to be discussed in QFT.

That these objects will enjoy Cartesian closure is not something you need to take care of, that's guaranteed by them being smooth sets.

I'd like to have a precise formulation of this theorem and what precisely is needed.

True, I should have provided this right away: Statement and pointers to the proofs are given here .

 

[Urs Schreiber]

I'd like to have a precise formulation of this theorem and what precisely is needed.

Statement and pointers to proofs are given at nLab:manifold structure of mapping spaces in this section.

 

[A. Neumaier]

Sure if you want to check full details from the foundations, you need to look at the full details starting from the foundations. But to just read the notes on QFT, you can take this as a black box. But the details are also easy to check, the category theory is minimal. It's really true that the theory of smooth sets is completely elementary certainly as compared to the heavy analytic machinery required for the usual infinite-dimensional manifold structures.

But do you get the same strong existence results as one gets for infinite-dimensional Lie groups, say? It is not without reason that the machinery is heavy!

Nah. You need as many smooth sets as you would otherwise consider smooth manifolds. One for spacetime, one for the field bundle, one for its jet bundle, one for the space of sections, one for the phase space. Just the obvious spaces that are to be discussed in QFT.

That these objects will enjoy Cartesian closure is not something you need to take care of, that's guaranteed by them being smooth sets.

What I meant is: To understand the concept of a smooth set you apparently need the whole category. Or can you define a smooth set without reference to a category, just as you can define a group without reference to a category?

 

[Urs Schreiber]

But do you get the same strong existence results as one gets for infinite-dimensional Lie groups, say?

The statement about the full embedding that I quoted means precisely that all standard theory embeds.

What I meant is: To understand the concept of a smooth set you apparently need the whole category.

Not in any non-trivial sense, no. A smooth sets is, as I mentioned, a declaration for each Cartesian space (abstract coordinate system) of what counts as a smooth function out of that space into the smooth set, subject to the evident condition that this is compatible with smooth functions between Cartesian space (abstract coordinate changes).

That''s it.

Or can you define a smooth set without reference to a category, just as you can define a group without reference to a category?

Sure. But this is a triviality unless you read some superficial scariness into the innocent word "category". When you define what a group is, you also want to know what counts as a homomorphism between two groups. The high-brow term for this is "the category of groups" but it means nothing else but "groups and maps between them".

 

[A. Neumaier]

A smooth sets is, as I mentioned, a declaration for each Cartesian space (abstract coordinate system) of what counts as a smooth function out of that space into the smooth set, subject to the evident condition that this is compatible with smooth functions between Cartesian space (abstract coordinate changes).

Well, I'd like to have a mathematically precise specification. Can I replace Cartesian space by $R^n$? Are there other significantly different Cartesian spaces that need to be catered for? Can one use fields like the p-adic numbers in place of the reals? (Some people are interested in p-adic physics!) Do the smooth functions have to be defined on all of $R^n$ or only on open subsets? What is the precise compatibility condition?

I wonder whether after all these things have been spelled out, the definition is really simpler than that of a manifold over a convenient vector space (in the sense of Kriegl and Michor), say.

I'd also like to be able to decide on the basis of the information provided questions such as whether any algebraic variety is a smooth set in an appropriate sense, or what restrictions apply.

When you define what a group is, you also want to know what counts as a homomorphism between two groups. The high-brow term for this is "the category of groups" but it means nothing else but "groups and maps between them".

But one can fist define what a group is, and then define what a homomorphism between groups is, and one does not the second concept to understand the first and to play with examples. I'd like to have a definition of smooth sets phrased in the same spirit. The categorial interpretation should be a second step that allows one to make certain universal constructions available, and not something already integrated into the definition.

 

[A. Neumaier]

Another quite popular non-perturbative approach is the renormalization-group approach ("Wetterich equation").

I guess, these more or less handwaving methods are not subject to the mathematically more rigorous approach discussed here, or can the here discussed approaches like pAQFT provide deeper insight to understand, why such methods are sometimes amazingly successful?

I believe that the Wetterich equation can be described on a reasonably rigorous level, though still with some uncontrolled approximations. But I haven't seen any concrete work in this direction by mathematical physicists.

 

[A. Neumaier]

By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here

• pAQFT 1: A first idea of quantum fields

In Section 2 of your nLab draft, you are unnecessarily onomatopoetic: ''expressions with repeated indicices''

 

[A. Neumaier]

The statement about the full embedding that I quoted means precisely that all standard theory embeds.

Apparently it means only that the standard examples are examples of the generalized concepts, not that the standard theorems also hold for the generalized concepts. I was asking for the latter. To get strong results for infinite-dimensional Lie groups, the latter probably need a full manifold structure, and not only a smooth set structure.

 

[A. Neumaier]

By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here

• pAQFT 1: A first idea of quantum fields

The title is somewhat misleading: 98% of the text is about classical field theory and only a few paragraphs at the end hint at quantum field theory through a sequence of remarks, without giving significant substance or interpretation. More appropriate would be something like ''The classical background needed for quantum field theory''.

 

[Urs Schreiber]

Well, I'd like to have a mathematically precise specification.

A detailed introduction is here: geometry of physics -- smooth sets . The quick way to state the definition is to say that a smooth set is a sheaf on the site whose objects are Cartesian spaces, whose morphisms are smooth functions between them, and whoe Grothendieck pre-topology is that coming from good open covers. But the introduction at geometry of physics -- smooth sets spells this out in elementary terms, not assuming any sheaf-theoretic background (or any other background except the concept of smooth functions between Cartesian spaces).

Can I replace Cartesian space by $R^n$?

Here "Cartesian space" means precisely : $\mathbb{R}^n$s.

Can one use fields like the p-adic numbers in place of the reals? (Some people are interested in p-adic physics!)

The analous definition work for any choice of test spaces with a concept of covering defined. If you take something like affinoid domains as in rigid analytic geometry you get somethng that deserves to be called "p-adic analytic sets" or the like. More relevant for physics is for instance the Choice of Stein spaces, in order to get "complex analytic sets". If you take affine schemes, you get ordinary algebraic spaces (among which ordinary schemes).

Do the smooth functions have to be defined on all of $R^n$ or only on open subsets? What is the precise compatibility condition?

One may equivalently take the site of open subsets of Cartesian spaces. Some authors do that. It does't change the resulting concept, though. The compatibility condition is gluing: The choice of what counts as a smooth function into your smooth set must be so that if you cover one Cartesian space by a set of other Cartesian spaces, then the smooth functions out of the former must be uniquely fixed by their restriction to those patches of the cover.

I'd also like to be able to decide on the basis of the information provided questions such as whether any algebraic variety is a smooth set in an appropriate sense, or what restrictions apply.

There is once you decide on what should count as a smooth function from a Cartesian space to the algebraic variety. In general there will not be a useful such choice, but if your algebraic variety happens to be complex-analytic, then of course there is, and you recover the underlying smooth manifold.

But one can fist define what a group is, and then define what a homomorphism between groups is, and one does not the second concept to understand the first and to play with examples.

Same for smooth sets. To recall, a smooth set is defined to be a choice, for each $n \in \mathbb{R}^n$ of a set, regarded as the set of smooth functions from $\mathbb{R}^n$ to the smooth set (called "plots"), such that this choice is compatible with smooth functions $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ and respects gluing, as above.

That's the definition. Next, a homomorphism between smooth sets is a map that takes these plots to each other, again respectiing the evident compositions.

 

[Urs Schreiber]

In Section 2 of your nLab draft, you are unnecessarily onomatopoetic: ''expressions with repeated indicices''

Thanks, fixed now.

 

[Urs Schreiber]

The title is somewhat misleading.

It keeps growing. In a few weeks the quantization will be covered, please have a little patience.

You see, this is my source from which I was going to successively produce Insights-articles here, one at a time. You are only seeing my master source only because the conversion to Insights-articles is running into technical difficulties.

 

[Urs Schreiber]

Apparently it means only that the standard examples are examples of the generalized concepts, not that the standard theorems also hold for the generalized concepts. I was asking for the latter. To get strong results for infinite-dimensional Lie groups, the latter probably need a full manifold structure, and not only a smooth set structure.

One needs full manifold structure for surprisingly few things. Everything that involves only differential forms instead of vector fields generalizes to all smooth sets. If vector fields get involved one needs to be careful, as for smooth sets which are not manifolds there appear different inequivalent concept of tangent spaces. But for applications to field theory, it turns out all one needs is differential forms on the spaces of field histories.

 

[A. Neumaier]

for smooth sets which are not manifolds there appear different inequivalent concept of tangent spaces. But for applications to field theory, it turns out all one needs is differential forms on the spaces of field histories.

I don't believe this. Already for classical existence proofs one needs to extend the infinitesimal description in terms of differential equations to a global description in terms of histories, and they are related as Lie algebras and Lie groups. In this relation the notion of tangent space is essential.

To be more concrete, can you show me a paper where Lie groups are studied from a smooth set point of view, and the typical difficulties in infinite dimensions are absent?

 

[Urs Schreiber]

I don't believe this. Already for classical existence proofs one needs to extend the infinitesimal description in terms of differential equations to a global description in terms of histories, and they are related as Lie algebras and Lie groups. In this relation the notion of tangent space is essential.

But not the tangent space to the space of histories.

To be more concrete, can you show me a paper where Lie groups are studied from a smooth set point of view, and the typical difficulties in infinite dimensions are absent?

The term to look for is "diffeological groups". For instance here

 

[Urs Schreiber]

Maybe I should re-amplify the point about diffeology:

The concept of "smooth sets" subsumes that of diffeological spaces , and essentially all examples of relevance in field theory fall in the class of diffeological space.

The diffeological spaces are the "concrete smooth sets". So the concept of diffeological spaces is a generalization of that of smooth manifolds, and the concept of smooth sets is yet a further generalization of diffeological spaces.

As far as the formalism is concerned, it is no harder to work in the generality of smooth sets than it is to work in the intermediate generality of diffeological spaces. But essentially all examples of smooth sets that appear in the context of field theory are actually diffeological spaces, and so if you are looking for literature on the subject, you should look for the keywords "diffeological spaces".

In particular, to highlight this once more, there is a down-to-earth non-categorical completely introductory and detailed textbook introducing all the standard material of differential geometry in terms of diffeological spaces.This is

• Patrick Iglesias-Zemmour:
  Diffeology
  Mathematical Surveys and Monographs
  Volume: 185; AMS 2013;

I don't think that studying this is necessary for following my notes, since the basic idea is really simple and really close to how physicists think anyway, but to all readers who do want to dig deeper into this differential geometric background to the theory I recommend looking at this textbook.

 

[dextercioby]

Urs, can you write in no more than 2 lines the connection (if any) between diffeologic spaces and topological spaces?

 

[Urs Schreiber]

Urs, can you write in no more than 2 lines the connection (if any) between diffeologic spaces and topological spaces?

Good question.

1. A topological space is a set $X$ equipped with information which functions $\mathbb{R}^n \longrightarrow X$ are continuous.
2. A diffeological space is a set $X$ equipped with information which functions $\mathbb{R}^n \longrightarrow X$ are smooth.

A leisurely exposition of the grand idea behind this is at motivation for sheaves, cohomology and higher stacks.

 

[DrDu]

It is since the number of loops counts the powers of $hbar$. This is clear from the path-integral formalism since you can understand the Dyson series also as saddle-point approximation of the path integral. See, e.g., Sect. 4.6.6 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

Yes, but the free theory which forms the starting point of the perturbation expansion contains already quantized electrons, photons, etc. How comes we consider this to be a classical theory?

 

[A. Neumaier]

Yes, but the free theory which forms the starting point of the perturbation expansion contains already quantized electrons, photons, etc. How comes we consider this to be a classical theory?

In an $\hbar$ expansion, the theory is expanded around the interacting classical limit, not around a free quantum field theory.

 

[DrDu]

In an $\hbar$ expansion, the theroy is expanded around the interacting classical limit, not around a free quantum field theory.

I know what you mean but I don't see how this fits with my naive view of e.g. QED where, when describing the scattering of, say, an electron from a photon. I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.

 

[vanhees71]

The point is that you can get the Klein-Nishina formula for Compton scattering, i.e., in the modern way by just evaluating the tree-level Feynman diagrams, by investigating scattering of a classical electromagnetic wave on an electron using the Dirac equation of the electron. The same holds for the photoeffect. You come quite far with the semiclassical approximation in QED, i.e., quantizing only the electron and keep the em. field as classical. The most simple argument for the necessity to quantize also the em. field is the existence of spontaneous emission, which afaik cannot be derived from the semiclassical theory.

 

[DrDu]

But you need a quantized electron, or is it sufficient to use a classical Grassmann valued field for the electron?