# Mathematical Quantum Field Theory - Fields - Comments

• Insights
Gold Member
Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Fields Continue reading the Original PF Insights Post.

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• fresh_42, Drakkith and Greg Bernhardt

Stephen Tashi
A perhaps naive conceptual question:

I think of a "history" of "events" in space time as set of things that actually happened - as you said:

A field history on a given spacetime Σ is a quantity assigned to each point of spacetime (each event), such that this assignment varies smoothly with spacetime points.

By contrast, I think of the definition of a field as involving events that might (or might not) have happened. You wrote:

For instance an electromagnetic field history (example 3.5 below) is at each point of spacetime a collection of vectors that encode the direction in which a charged particle passing through that point will feel a force (the “Lorentz force“, see example 3.5 below).

So I don't understand the wording "will feel" unless the subject matter we are considering is "all possible histories in space time" - something like a "many worlds" point of view - if not "many worlds" for the spacetime of entire universe, at least a "many labs" point of view for some given type of experiment. From that viewpoint, a field history describes a set of different possible physical situations, each of which is considered to be an example of "the same" field history. (By analogy, in classical physics, "the" electric field of a unit positive charge located at (0,0,0) is not a description of one particular physical situation. Instead, it describes a general type of situation that can, in principle, be set up in different laboratories using different points in space as (0,0,0).)

A simplistic model is that, in a given universe or experiment, a particle either definitely did or or did not pass the given point at the given time. So we can only talk about what force a particle "would have felt" by considering the given experiment to be one experiment in a set of experiments of the same general type. ( That won't disturb physicists, but it might worry logicians since statements of the form "If particle W passed through point P then ... such-and-such" are all true when particle W didn't pass through point P. )

Is the simplistic model satisfactory? Or must we discard the notion that a particle has a definite position at a given time right at the outset?

Gold Member
I don't understand the wording "will feel"

Given an EM field history and a trajectory of an electron, then there is a Lorentz force.

Maybe I might change "will" to "would", if that helps?

The connotation of "many worlds" is not appropriate here, it's rather about possible worlds. Maybe it would help if I say "space of possible field histories"? (If you care about the logic of possibility, the right framework is type theory and specifically modal type theory. I have some exposition of this with an eye towards physics in Modern Physics formalized in Modal Type Theory. But this is esoteric, not for the faint hearted; I am just mentioning it in case you do want to dig deep into the concept of modality in physics.)

It makes sense and is necessary to speak, for any type of fields, of what qualifies as a field history of that type, before asking whether that field history is realizable in nature and before asking whether it is realized in the observed universe.

There are these stages of conceptualization:

• the type of field ##E##: what type of quantity gets assigned to a spacetime point;
• a field history ##\Phi## of that type, hence an assignment of such quantities to spacetime point (type theorists call this a "term" of that type);
• the space ##\Gamma_\Sigma(E)## of all possible field histories of that type (type theorists call this a function type);
• the subspace ##\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}## of on-shell field histories, those that obey the prescribed equation of motion (the laws of nature, if you like; notice that there is not one fixed choice for these), type theorists call this a mere proposition (i.e. the proposition "The field history ##\Phi## solves the equations of motion.")

Part of your question might be read as asking if we could not just consider the last item the on-shell space of field histories, without considering also the larger space of possible field theories that it is a sub-space of. It is indeed true that one can do this, and often does. It is a specific property of what is called Lagrangian field theory that we obtain this space (and its presymplectic structure) in such a sequence of steps as above. One of the deep mysteries of our world is that most field theories of interest are Lagrangian field theories (and many of those which are not, such as the chiral WZW model, are duals of those that are).

vanhees71
Gold Member
Well, I'm a bit uncertain about this definition of the field too. It's pretty much a naive classical picture, expressed in mathematical formal terms. Indeed, a good lecture on classical electromagnetism starts with the operational definition of the electromagnetic field via its action on charged bodies (idealized in a naive way to "point charges") in terms of the Lorentz force. Now it is pretty clear that there is no consistent classical many-body theory of point charges due to the notorious radiation-reaction problem, which is only solved approximately (fortunately sufficient for all practical purposes, where it's needed to, e.g., construct particle accelerators like the LHC).

The best theory we have so far is QFT, and there you usually have just the S-matrix elements (leading to transition probabilities for a given asymptotic free in state to a given asymptotic free out state) or some macroscopic bulk properties of many-body systems.

Gold Member

So it's good that we are running this series then! Lots of basics of QFT are widely unknown.

The field operators of QFT are observables on the fields as defined here, hence functionals on the space of fields as defined here. We get to that in chapter 7.

• bhobba
vanhees71
Gold Member
Well, I'm only a naive theoretical physicist, but I think that point of view doesn't make sense. The field operators cannot, in general, represent local observables (even if they are self-adjoint). There are several reasons. One is that only gauge-invariant properties are observable, and the operator of the electromagnetic field ##\hat{A}^{\mu}## is not gauge invariant.

Another even more fundamental example are fermionic operators like a Dirac-field operator ##\hat{\psi}_a(x)## (where ##a## is an index counting spinor components). From the canonical field-anticommutator relations, it's clear that the fields do not commute with space-like separated space-time arguments, which should be the case to ensure microcausality, which is the way to ensure the unitarity and Poincare invariance of S-matrix elements, as well as the Linked-Cluster Property (see Weinberg, QT of Fields Vol. 1).

Gold Member
that point of view doesn't make sense.

This is not a point of view, but the very definition of quantum theory: Quantum operators are functions on the phase space (equipped with a non-commutative product operation), and the phase space is the space of solutions of the equations of motion, and these equations of motion are imposed on the fields, and these are sections of the field bundle.

If you are impatient waiting for the series to arrive at the quantum operators in a few chapters, I can recommend Rejzner 16 for a textbook account on QFT that leaves no mystery about the concepts.

The field operators cannot, in general, represent local observables (even if they are self-adjoint). There are several reasons. One is that only gauge-invariant properties are observable, and the operator of the electromagnetic field ##\hat{A}^{\mu}## is not gauge invariant.

There is no contradiction here. The gauge invariant observables are built from gauge invariant combinations of the field operators.

A general observable is a smooth functional

$$A \;:\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \longrightarrow \mathbb{C}$$

on the space of on shell field histories (the covariant phase space). Among these are the linear ones, these are the distributions. Among those are the delta-distributions, namely the point evaluation observables, known as the field observables ##\mathbf{\Phi}^a(x)##, defined by sending a field history ##\Phi \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}## to the value ##\Phi^a(x)## of its ##a##-component at spacetime point ##x##. In terms of these all other observables are expressed by smearing, convolution and taking products.

Another even more fundamental example are fermionic operators like a Dirac-field operator ##\hat{\psi}_a(x)## (where ##a## is an index counting spinor components). From the canonical field-anticommutator relations, it's clear that the fields do not commute with space-like separated space-time arguments, which should be the case to ensure microcausality, which is the way to ensure the unitarity and Poincare invariance of S-matrix elements, as well as the Linked-Cluster Property (see Weinberg, QT of Fields Vol. 1).

Right, but again there is no contradiction here. This is why it is important to understand that fermionic fields are odd-graded elements in a super-algebra. This in particular means that while odd in themselves (in particular anti-commuting) they become even when regarded in odd-parameterized families. The present chapter "3. Fields" lays the groundwork for the discussion of this important point in its section 4 on supergeometry.

vanhees71
Gold Member
Well, I guess it's a problem of terminology. Nowadays there seems to be nearly no overlap between mathematical and theoretical physicists anymore. The language of both groups are so different that misunderstandings are almost predetermined. This is really a pity since a theoretical physicist like me lacks the rigor of the mathematical physicst, while the latter often forgets the physics background of the theory.

My only point was that you claimed the field operators represent observables, but that's not true. To represent observables, they must fulfill certain constraints to make sense as such. Of course, all the operators representing (local or global) observables are built by the fundamental field operators, whose properties are constructed via the various physically relevant representations of the proper orthochronous Lorentz group.

Gold Member
Well, I guess it's a problem of terminology.

I had read your comment in #4 as doubting the point of the space of field histories on the grounds that this looks to you like "naive classical" field theory as opposed to be proper quantum field theory.

In reaction I tried to point out that the proper quantum field theory, say in terms of the S-matrix that you mentioned, is embodied by quantum observables which are indeed functionals on this space of "classical naive" field histories.

• dextercioby and vanhees71
vanhees71
Gold Member
Ok, as I guessed we mean the same thing but use different terminology :-)).

• bhobba
Gold Member
Ok, as I guessed we mean the same thing but use different terminology :-)).

Thanks for all your feedback, I value that a lot.

My ambition is to discuss the standard QFT theory in standard terminology, just augmented by whatever it takes to make it clear and precise. The issue we are facing here is that the word "field" is traditionally used in an ambiguous way. Therefore the chapter "3 Fields" of the series splits it up into the three different meaning it has:

1. type of fields (or "field species") made clear and precise by the field bundle,
2. field histories, made clear and precise by the sections of the field bundle,
3. field observables, made clear and precise by the smooth functions on the space of field histories.

(With some qualifiers omitted here that don't affect the general point, i.e. eventually we restrict to the observables that are both on-shell as well as gauge invariant, namely to the cohomology of the BV-BRST differential acting on the graded space enhancement of these observables. )

I suspect that maybe you may have wanted me to say "classical field" where I say "field history" (?), but I won't do that, because the distinction between 2. field histories and 3. field observables exists in classical field theory just as well.

It is a curious fact that maybe remains underappreciated (?) that the "quantum field observables" or "quantum field operators" of quantum field theory are indeed functionals on the same space of (on-shell) field histories; what makes them "quantum" is not that the concept of field history changes, but just that the product on these functionals gets deformed.

This is just as in quantum mechanics: When we quantize the free particle in some space ##X##, we do not change the meaning of "smooth trajectory in ##X##" (which is a field history in this case), but on the algebra of functionals on this space of field histories (such as the functional "##x^\mu(t)##", the analog of ##\Phi^a(x)## in field theory, which send a field history to the value of its position at some point ##t## in its field history) we change the product -- namely from the pointwise product to the Heisenberg operator product.

vanhees71
Gold Member
Well, I'm obviously of the opposite opinion. Your example with the non-relativistic quantization (in the "1st quantization formalism") makes this very clear. It is important, in my opinion, to emphasize the quite radical difference between classical and quantum physics early on. So in this example it is important to understand that the classical description of the motion of particles in terms of trajectories in phase space has to be given up. The quantum state is not a point (or equivalently its trajectory under Hamiltonian motion) in phase space anymore but an equivalence class of preparation procedures, leading to probabilistic information about measurements of observables, formally given by the Statistical Operator of the system (or equivalently for pure states a unit ray in Hilbert space).

The classical fields are of course defined operationally either as local quantities like energy, momentum, angular-momentum, charge densities or in the case of entities like the electromagnetic fields by their action on matter (either formalized as point particles or, more "natural" in the field-theoretical context, continuum mechanical ways).

The quantum field theory case is again pretty different, particularly in the relativistic case. The fields provide a way to construct a Hilbert space appropriate for situations, where particle numbers are not conserved anymore, i.e., the Fock space to begin with, and that's possible only for free fields, which also provide a clear definition of particles as states of good occupation number 1. Observable in the sense of particles are thus only asymptotic free states, and thus the main physically relevant quantity in vacuum QFT are S-matrix elements or the corresponding cross sections, or decay rates (lifetimes) of "unstable particles".

From this point of view, it's rather unclear to me what you mean by "field history" in the quantum case. The fields are no longer directly observable and thus you cannot give a "field history" in the sense of observable facts about nature.

Gold Member
Well, I'm obviously of the opposite opinion.

I hope we can make this not be a matter of opinion, but of mathematical fact.

it's rather unclear to me what you mean by "field history" in the quantum case.

That's what precise definitions are for, to remove such ambiguity!

I maintain that to define the quantum observables of a Lagrangian field theory, you have to define them as functionals on the space of on-shell sections of the field bundle. This is not in contradiction with the fact that once these are equipped with their quantum operator product, the result is a non-commutative algebra from which alone the original space of field histories may not be recovered exactly. But to get to that point, we need to say exactly what that algebra of quantum observables is, and that does require the space of field histories.

The series will get to this point in a few chapters. Maybe we can pick up the discussion again then. I just like to amplify that nothing I am doing in the series is non-standard or controversial, it follows the established clear and precise formulation of quantum field theory.

Stephen Tashi
The connotation of "many worlds" is not appropriate here, it's rather about possible worlds.

Since QFT encompasses QM and there are different physical interpretations of QM, I don't expect QFT to have a unique physical interpretation. (If I'm wrong about that, please tell me.) What I would like to understand is where physical interpretation begins to become ambiguous in the exposition of QFT.

I associate a boldness with talking about "spacetime" because it suggests that one is really willing to talk about the entire universe. Perhaps, I shouldn't make that association. For example, if classical physics presents a formula for the electric field around a unit positive charge "in all of space", this can't be taken literally. It has to be prefaced by some remark like "Imagine that the only thing in the universe is a unit positive charge" ( i.e an "imaginable" world) or "Consider a vast region of space that is empty except for a positive charge" ( i.e. a finite subset of the actual world).

It makes sense and is necessary to speak, for any type of fields, of what qualifies as a field history of that type, before asking whether that field history is realizable in nature and before asking whether it is realized in the observed universe.

I understand that a field history can exist as a mathematical concept -i.e. that one can specify a formula that associates a quantity with each 4-tuple of real numbers. When we are talking about realizable field histories, a (perhaps ridiculous) question can be asked: "If H1 and H2 are distinct realized field histories, can they refer to the same physical quantity?". I think the correct answer is "Yes" because we don't take the realized "spacetime" literally. For example, if both field histories refer to physical property P, they can be regarded as approximate descriptions of two different experiments on P conducted in different laboratories at different times. So the "spacetime" of H1 isn't really all of space and time.

In mathematics, one can distinguish between a mathematical object of one type (e.g. a group) and a mathematical object of another type that talks about that object "applied to" another mathematical object (e.g. a group action on a set). In mathematical physics, I don't detect any tradition of formalizing the division between a mathematical object and its application to the actual world. (For example, in texts on group theory applied to chemistry, what is called a "group" sometimes morphs into a "group action" without any warning to the reader that a fundamental boundary has been crossed.) No exposition of mathematical physics ought to be critcized for not formalizing a distinction between the The Mathematical and the The Actual. I'm just curious if QFT might take the unusual step of of doing that.

Maybe it would help if I say "space of possible field histories"? (If you care about the logic of possibility, the right framework is type theory and specifically modal type theory. I have some exposition of this with an eye towards physics in Modern Physics formalized in Modal Type Theory. But this is esoteric, not for the faint hearted; I am just mentioning it in case you do want to dig deep into the concept of modality in physics.)

I agree that one can formalize the concept of "possibility" in the sense that one can create a formal language that employs a mathematical concept called "possibility" and show how statements in that language imply other formal statements - and how these statements can be matched up with "natural language" statements about possibility. Perhaps that's the best approach.

Among the concepts of "Actual" , "Possible", "Probable", the concept of "Actual" seems the clearest. A result of a scientific experiment is "Actual". Perhaps "Possible" and "Probable" can't be defined in terms of "Actual".

Gold Member
In mathematical physics, I don't detect any tradition of formalizing the division between a mathematical object and its application to the actual world.

There is such a tradition in the philosophy of physics. The technical term there for this division, or rather for the relation between the two is "coordination".

No exposition of mathematical physics ought to be critcized for not formalizing a distinction between the The Mathematical and the The Actual. I'm just curious if QFT might take the unusual step of of doing that.

Typical QFT texts do not, but with the concept of field developed with due care, via field bundles, sections and functions on the space of sections, it is at least straightforward to get into the discussion of "coordination".

vanhees71
Gold Member
I hope we can make this not be a matter of opinion, but of mathematical fact.

That's what precise definitions are for, to remove such ambiguity!

I maintain that to define the quantum observables of a Lagrangian field theory, you have to define them as functionals on the space of on-shell sections of the field bundle. This is not in contradiction with the fact that once these are equipped with their quantum operator product, the result is a non-commutative algebra from which alone the original space of field histories may not be recovered exactly. But to get to that point, we need to say exactly what that algebra of quantum observables is, and that does require the space of field histories.

The series will get to this point in a few chapters. Maybe we can pick up the discussion again then. I just like to amplify that nothing I am doing in the series is non-standard or controversial, it follows the established clear and precise formulation of quantum field theory.
Ok, but your definition via fiber/jet bundles so far is about classical field theory, right? Then I can understand it (at least in an intuitive way, translating the mathematical formalism to my naive understanding of field theory). On the quantum level a "history of interacting fields" is at least problematic, i.e., the physical interpretation of "transient states" is not at all clear in standard theoretical physics. Since you say "on-shell sections of the field bundle", I can imagine that your approach is formalizing the naive theoretical physics "definitions" of asymptotic free states, and then you have a (naive) particle interpretation, although there are also problems left at least in QED.

Gold Member
your definition via fiber/jet bundles so far is about classical field theory, right?

As I said before, the quantum operators arise as functionals on this space of on-shell sections of the field bundle. This will be the topic of chapter 7 and following, going public in a few days from now. Let's pick up the discussion then.

• vanhees71
vanhees71
Gold Member
Ok. I'm too impatient :-)).

• Urs Schreiber
Demystifier
Gold Member
One is that only gauge-invariant properties are observable, and the operator of the electromagnetic field ##\hat{A}^{\mu}## is not gauge invariant.
But to make any computation in quantum theory at all, one must first fix the gauge. And if the gauge is fixed completely (like in Coulomb gauge), then there is a one-to-one correspondence between ##A^{\mu}## and ##F^{\mu\nu}##. So when the gauge is fixed, then ##A^{\mu}## is an observable.

In fact, saying that ##A^{\mu}## is not an observable in QED due to gauge invariance is like saying that the position ##x^i## is not an observable in QM due to translation and rotation invariance. Once the gauge (or spacial coordinates) is fixed, the ##A^{\mu}## (or ##x^i##) becomes an observable.

Demystifier
Gold Member
the phase space is the space of solutions of the equations of motion
The only problematic word here is "is". The phase space is the space of initial conditions of the equations of motion. Initial conditions are not solutions. However, there is a one-to-one correspondence between initial conditions and solutions. So a more correct statement would be that phase space is in one-to-one correspondence with the space of solutions of the equations of motion.

Of course, mathematicians like to think that when two objects are in one-to-one correspondence, then they are, in a certain sense, "the same". But in many senses they are not the same. For instance, just because a one-to-one correspondence exists doesn't mean that this correspondence is known. (Just because the solution for given initial conditions exists doesn't mean that this solution is known.) So if two objects are in one-to-one correspondence but one is known and the other is unknown, it can be very confusing to think of the two objects as being "the same".

Another example: Consider the logical operation NOT x, where x is either logical 0 or logical 1. Clearly, NOT x is in one-to-one correspondence with x. However, no logician will say that NOT x is the same as x.

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• strangerep
Gold Member
But to make any computation in quantum theory at all, one must first fix the gauge. And if the gauge is fixed completely (like in Coulomb gauge), then there is a one-to-one correspondence between ##A^{\mu}## and ##F^{\mu\nu}##. So when the gauge is fixed, then ##A^{\mu}## is an observable.

That's right, that's the content of the upcoming chapter 12.

• Demystifier
Gold Member
Of course, mathematicians like to think that

I like to caution against this habit of organizing people into camps and then declaring what they do and do not think. I keep hearing what a) Mathematicians, b) Physicists, c) Mathematical physcists etc. allegedly a) think, b) do or don't understand, and c) do or do not care about.

Irrespective of the at best shaky truth of these statements and of the curious assumption of universal intellecutal laziness suggested thereby, this is a perspective inappropriate for the beautiful unity of the quest for truth. Just like true faith is not actually helped by organizing people into Catholic, Presbyterians etc. so true insight is not helped by behaving as if the bureaucratic organization of the academic system is something that researchers are unable to transcend.

That said, the details of phase spaces, and the subtle but important distinction between phase spaces associated with a Cauchy surface and the "covariant" phase space of all solutions is going to be the content of chapter 8, which comes online next week, I suppose.

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• dextercioby and Demystifier
Demystifier
Gold Member
This is not a point of view, but the very definition of quantum theory: Quantum operators are functions on the phase space (equipped with a non-commutative product operation), and the phase space is the space of solutions of the equations of motion, and these equations of motion are imposed on the fields, and these are sections of the field bundle.
Is the above valid for the spin operator? If so, what is the corresponding phase space and what are the equations of motion?

vanhees71
Gold Member
But to make any computation in quantum theory at all, one must first fix the gauge. And if the gauge is fixed completely (like in Coulomb gauge), then there is a one-to-one correspondence between ##A^{\mu}## and ##F^{\mu\nu}##. So when the gauge is fixed, then ##A^{\mu}## is an observable.

In fact, saying that ##A^{\mu}## is not an observable in QED due to gauge invariance is like saying that the position ##x^i## is not an observable in QM due to translation and rotation invariance. Once the gauge (or spacial coordinates) is fixed, the ##A^{\mu}## (or ##x^i##) becomes an observable.
There is a difference between global symmetries and local symmetries. While ##A^{\mu}## contains unphysical degrees of freedom, which don't do anything in the properly formulated theory because of gauge invariance (i.e., all the unphysical degrees of freedom are cancelled for observable on-shell S-matrix elements as long as the calculation obeys the local gauge symmetry and the corresponding Ward-Takahashi identities of the Green's functions), the position operators (if they exist, which is the case for all massive particles and for massless particles with spin 0 or 1/2) are gauge invariant and thus bona fide representatives for observables.

vanhees71