I Confusion about Scattering in Quantum Electrodynamics

[A. Neumaier]

No, in fact that's the point. Time evolution in QED is not unitary, but a contractive Markovian process. This is at the non-perturbative level. Perturbatively time evolution is unitary.

Please give a reference for this strange claim.

 

[LittleSchwinger]

Please give a reference for this strange claim.

A nice worked model is here, done in the non-relativistic limit:
https://arxiv.org/abs/2212.02599

The general theory is covered in Frohlich's proceeding papers, which include references to Buchholz's previous works on the topic. I prefer this because the other papers are "advanced" mathematical physics. I think most would find this easier to follow.

Roughly speaking due to a contraction of the algebra of observables we have a time evolution that is most likely unitary on the observables, but non-unitary/Markovian on the states.

 

[A. Neumaier]

A nice worked model is here, done in the non-relativistic limit:
https://arxiv.org/abs/2212.02599

The general theory is covered in Frohlich's proceeding papers, which include references to Buchholz's previous works on the topic. I prefer this because the other papers are "advanced" mathematical physics. I think most would find this easier to follow.

Roughly speaking due to a contraction of the algebra of observables we have a time evolution that is most likely unitary on the observables, but non-unitary/Markovian on the states.

However, this is not QED but a nonrelativistic approximations. Approximations often violate unitarity, due to neglect of degrees of freedom that can absorb energy or entropy.

 

[physwiz222]

This is a very reasonable question that I had puzzled over for a long time before understanding what actually happens.

In textbook QED one starts with a Fock space and finds out that all quantities of interest are ill-defined without renormalization. By Haag's theorem, this is necessarily so for relativistic QFTs based on the interaction picture (which is needed for perturbation theory) - the latter simply does not exist. Thus one introduces renormalization, which works smoothly only at the level of the S-matrix. There is a widespread view that in relativistic QFTs, only the asymptotic descriptions (i.e., the S-matrix) makes quantitative sense, and can be computed using renormalization, together with IR regularization via cumulative cross sections or coherent states.

The reason is the difficulty to give QED (and other physically relevant relativistic QFTs) a rigorous mathematical basis: In a logically impeccable sense (in principle being able to produce model predictions to infinite accuracy) of a single interacting relativistic QFT in 4 spacetime dimensions is known. One of the Clay Millenium Problems asks for a construction for 4D Yang-Mills, which is though to be the simplest relativistic QFT in 4 spacetime dimensions potentially in reach by current methods.

On the other hand, suppose a mathematical construction of QED (or any relativistic QFT in 4 spacetime dimensions) exists. This would imply a realization of the Wightman axioms. Any such realization produces a Hilbert space with a unitary representation of the Poincare group, and the generator $H=P_0$ of the time translation acts as a Hamiltonian producing the temporal evolution of the states.

Currently only free fields (and their quasi-free cousons) - such as QED with zero electron charge - are known to realize the Wightman axioms in 4 spacetime dimensions. Their states indeed have an explicit temporal evolution, given by the traditional free Hamiltonian on Fock space.

Thus the reason why textbooks are silent about the temporal evolution is because theory is not developed enough to be able to say more than trivialities about it.

I did some research on this issue and it seems if my understanding is right the fundamental issue to describing the finite time dynamics of Interacting Fields is Haag’s Theorem because it states that the Interacting States cant be described by a combination of Fock States.

Thus there is no way to describe how the occupation numbers evolve and how the states morph as the Interacting states are not describable in terms of the Fock States at finite time so at least for finite time prohibits the idea of Perturbation theory and series expansions in general.

Its akin to how a combination of plane waves with spatial variation only in the x direction no matter how many cant describe a shape like a sphere as its not describable in terms of solely plane waves with variations in the x directions as it also varies along y and z. Interacting states are similar to this.

 

[A. Neumaier]

I did some research on this issue and it seems if my understanding is right the fundamental issue to describing the finite time dynamics of Interacting Fields is Haag’s Theorem because it states that the Interacting States cant be described by a combination of Fock States.

Thus there is no way to describe how the occupation numbers evolve and how the states morph as the Interacting states are not describable in terms of the Fock States

Your argument is not conclusive as there is no necessity for the Hilbert space of a quantum field theory to be a Fock space. It only follows that the Fock space concept is inadequate to capture interacting QFT.

Note that there are many rigorous constructions of interacting QFTs in 2 and 3 spacetime dimensions, in spite of the validity of Haag's theorem there. But in no case is the finite-time dynamics given in a Fock space.

 

[physwiz222]

Your argument is not conclusive as there is no necessity for the Hilbert space of a quantum field theory to be a Fock space. It only follows that the Fock space concept is inadequate to capture interacting QFT.

Note that there are many rigorous constructions of interacting QFTs in 2 and 3 spacetime dimensions, in spite of the validity of Haag's theorem there. But in no case is the finite-time dynamics given in a Fock space.

Well what I said was that the reason no one computes a time dependent state for interacting QFT by using the fock basis to construct a state pf the form c0(t)|0>+c1(t)|k> is because the Interacting states arent describable by Fock states.
My argument was that Haag’s Theorem effectively prohibits constructing a time dependent approximate state using the Fock basis.

 

[Morbert]

However, this is not QED but a nonrelativistic approximations. Approximations often violate unitarity, due to neglect of degrees of freedom that can absorb energy or entropy.

He presents a relativistic account here which involves the same theorem by Buchholz. (see equation 7)

 

[A. Neumaier]

He presents a relativistic account here which involves the same theorem by Buchholz. (see equation 7)

But there Fröhlich doesn't make the strange claim that

Time evolution in QED is not unitary, but a contractive Markovian process.

Indeed, in a very recent paper, Buchholz and Fredenhagen explicitly discuss the loss of information entailed in constructing an arrow of time for QED. The contractivity is explained by an approximation that restricts to what an observer can know.

 

[Morbert]

Indeed, in a very recent paper, Buchholz and Fredenhagen explicitly discuss the loss of information entailed in constructing an arrow of time for QED. The contractivity is explained by an approximation that restricts to what an observer can know.

I can see how using loss of access to information to construct a family of event algebras that satisfies a contractivity is in a sense an approximation. On the other hand, Fröhlich makes an implicit distinction between quantum theory and a quantum theory of events. Perhaps a quantum theory of of events (whether the theory is quantum-mechanical or quantum-electrodynamic) cannot otherwise be constructed.

 

[A. Neumaier]

using loss of access to information to construct a family of event algebras that satisfies a contractivity is in a sense an approximation.

This is the general mechanism; nothing is special here. See any derivation of decoherence or Lindblad master equations:

Restricting attention to less than complete information (absolutely necessary for an observer since complete information is never available) always turns the exact unitary evolution into a contractive evolution.

 

[Morbert]

This is the general mechanism; nothing is special here. See any derivation of decoherence or Lindblad master equations:

Restricting attention to less than complete information (absolutely necessary for an observer since complete information is never available) always turns the exact unitary evolution into a contractive evolution.

But is this general framing (exact unitary evolution, restriction of attention, and approximate contractive evolution) necessary in algebraic QFT? Fröhlich implies the reverse in some of his writing. In his article on the time-evolution of states He says unitary evolution and nonlinear state collapse "can be understood as an approximation to the fundamental [stochastic] law of evolution of states". I.e. in his AQFT formalism, stochastic state evolution is not arrived at by first sketching some unitary evolution and then coarsening it. He instead seems to start with a principle like Huygen's principle and event algebras implied by it.

 

[A. Neumaier]

But is this general framing (exact unitary evolution, restriction of attention, and approximate contractive evolution) necessary in algebraic QFT? Fröhlich implies the reverse in some of his writing. In his article on the time-evolution of states He says unitary evolution and nonlinear state collapse "can be understood as an approximation to the fundamental [stochastic] law of evolution of states". I.e. in his AQFT formalism, stochastic state evolution is not arrived at by first sketching some unitary evolution and then coarsening it. He instead seems to start with a principle like Huygen's principle and event algebras implied by it.

Well, one can always start anywhere, and build from there.

By definition, QED starts from unitary evolution. If others start at an approximation of QED, it is their freedom. But it doesn't change the fact that the unitary and deterministic evolution of the states is the fundamental one from which the stochastic approximation can be derived. One cannot derive the unitary evolution of QED from its stochastic conic approximation.

 

[WernerQH]

By definition, QED starts from unitary evolution.

Perhaps that's why we can't find a mathematically satisfactory formulation of QED, starting from this "definition".

 

[A. Neumaier]

Perhaps that's why we can't find a mathematically satisfactory formulation of QED, starting from this "definition".

The definition works constructively in 3 spacetime dimensions, and there are no known mathematical reasons (only speculations) why this shouldn't extend to 4 dimensions.

Note also that Fröhlich's approach doesn't construct the contractive variant of QED he is speculating about! Thus your argument also applies to his approach.

 

[WernerQH]

there are no known mathematical reasons (only speculations) why this shouldn't extend to 4 dimensions.

Keep searching. Good luck!

 

[physwiz222]

Your argument is not conclusive as there is no necessity for the Hilbert space of a quantum field theory to be a Fock space. It only follows that the Fock space concept is inadequate to capture interacting QFT.

Note that there are many rigorous constructions of interacting QFTs in 2 and 3 spacetime dimensions, in spite of the validity of Haag's theorem there. But in no case is the finite-time dynamics given in a Fock space.

Has anyone at least attempted naively applying Perturbation Theory to QFT at finite times. I know it wont work because of Haag’s Theorem as the states arent describable as particle states even infinitely many.

What I am asking now is has anyone at least tried to apply Perturbation Theory at finite times by computing the Transition Amplitudes between states at finite time to construct superpositions of the form c0(t)|0>+ck(t)|k>+…cn(t)|n> just like in standard QM.

I know this doesnt work but has anyone at least tried this naive approach and ran into problems. If so could u link some resources like Papers where this approach was tried and failed.

 

[A. Neumaier]

Has anyone at least attempted naively applying Perturbation Theory to QFT at finite times. I know it wont work because of Haag’s Theorem as the states arent describable as particle states even infinitely many.

What I am asking now is has anyone at least tried to apply Perturbation Theory at finite times by computing the Transition Amplitudes between states at finite time to construct superpositions of the form c0(t)|0>+ck(t)|k>+…cn(t)|n> just like in standard QM.

I know this doesnt work but has anyone at least tried this naive approach and ran into problems. If so could u link some resources like Papers where this approach was tried and failed.

This naive approach was tried very early in the history of QFT and discarded since it led to UV divergences; see, e.g., Wikipedia.

Haag's theorem showed rigorously that this naive approach cannot work, i.e., why renormalization is unavoidable if one starts with Fock space.

 

[AndreasC]

This naive approach was tried very early in the history of QFT and discarded since it led to UV divergences; see, e.g., Wikipedia.

Haag's theorem showed rigorously that this naive approach cannot work, i.e., why renormalization is unavoidable.

But wait, I thought divergences related to Haag's theorem are IR, not UV!

 

[A. Neumaier]

But wait, I thought divergences related to Haag's theorem are IR, not UV!

IR divergences appear only in theories containing a massless field (such as QED). But Haag's theorem applies to all interacting fields, hence also to massive scalar $\Phi^4$ QFT, which has no IR divergences.

The problem exposed by Haag's theorem are the superselection sectors that appear when the infinite volume limit is taken. Though the latter is a long-distance (IR) phenomenon, the associated divergence can come from both UV and IR aspects.

 

[A. Neumaier]

IR divergences appear only in theories containing a massless field (such as QED). But Haag's theorem applies to all interacting fields, hence also to massive scalar $\Phi^4$ QFT, which has no IR divergences.

The problem exposed by Haag's theorem are the superselection sectors that appear when the infinite volume limit is taken. Though the latter is a long-distance (IR) phenomenon, the associated divergence can come from both UV and IR aspects.

See also https://physics.stackexchange.com/questions/778763

 

[AndreasC]

The problem exposed by Haag's theorem are the superselection sectors that appear when the infinite volume limit is taken. Though the latter is a long-distance (IR) phenomenon, the associated divergence can come from both UV and IR aspects.

Ah ok this is what I had in mind, thanks for the clarification.

 

[vanhees71]

Also the IR divergences are dealt with, at least in gauge theories, by using the correct asymptotic states, which also correct the naive "naked-particle picture" of the "charged" particles in such a theory. The true asymptotic states of, e.g., an electron in QED is an "infraparticle state", which describes the "bare electron + it's own electromagnetic field around it" (speaking in a hand-waving heuristic way, of course).

What's done FAPP is anyway to use some regularization procedure. To get rid of the formal trouble with the non-existence of the interaction picture a la Haag, you can introduce a quantization volume, e.g., a cube imposing periodic spatial boundary conditions on the fields and then taking the infinite-volume limit only at the end.

 

[physwiz222]

This naive approach was tried very early in the history of QFT and discarded since it led to UV divergences; see, e.g., Wikipedia.

Haag's theorem showed rigorously that this naive approach cannot work, i.e., why renormalization is unavoidable if one starts with Fock space.

Why cant Renormalization then be used to remove the divergences and make this naive approach work then. You say this naive approach fails due to divergences. Isnt that what Renormalization was invented for to fix such divergences. Is QFT at finite times not renormalizable is that why Renormalization cant make this naive approach work.

 

[A. Neumaier]

Why cant Renormalization then be used to remove the divergences and make this naive approach work then. You say this naive approach fails due to divergences. Isnt that what Renormalization was invented for to fix such divergences. Is QFT at finite times not renormalizable is that why Renormalization cant make this naive approach work.

Textbook renormalization is done only at the asymptotic level of the S-matrix (involving the limit $t\to\pm\infty$) , where it works for massive theories (only). The reason is that the asymptotic states of a massive QFT are free particle states, hence describable by a free QFT in a Fock space.

For theories with massless particles, the asymptotic states of a QFT are no longer free particle states but infraparticle states. Therefore the asymptotic state space defining the S-matrix is not a Fock space, and infrared problems are not cured by renormalization, not even at asymptotic times. They must be handled differently, conventionally by coherent state methods (in addition to renormalization, which handle the UV issues); see post #34. Very recently, Duch and Dybalski wrote a survey paper about what is known about curing the IR problem at a more rigorous level.

Renormalization at finite times can currently be done with the CTP (closed time path) formalism; though not for few-particle states, but only for states describing the hydrodynamic or kinetic many-body regime. See post #59.

 

[physwiz222]

Textbook renormalization is done only at the asymptotic level of the S-matrix (involving the limit $t\to\pm\infty$) , where it works for massive theories (only). The reason is that the asymptotic states of a massive QFT are free particle states, hence describable by a free QFT in a Fock space.

For theories with massless particles, the asymptotic states of a QFT are no longer free particle states but infraparticle states. Therefore the asymptotic state space defining the S-matrix is not a Fock space, and infrared problems are not cured by renormalization, not even at asymptotic times. They must be handled differently, conventionally by coherent state methods; see post #34. Very recently, Duch and Dybalski wrote a survey paper about what is known about curing the IR problem at a more rigorous level.

Renormalization at finite times can currently be done with the CTP (closed time path) formalism; though not for few-particle states, but only for states describing the hydrodynamic or kinetic many-body regime. See post #59.

So essentially the reason this naive approach fails is a combination of divergences which we dont know how to deal with at finite time and Haag’s Theorem prohibiting describing the interacting states as superpositions. With all this in mind just out of curiosity is Relativistic QED at finite times nonrenormalizable or we simply dont know how to apply renormalization.

 

[A. Neumaier]

So essentially the reason this naive approach fails is a combination of divergences which we dont know how to deal with at finite time and Haag’s Theorem prohibiting describing the interacting states as superpositions.

Haag's theorem also shows what is needed to deal with at finite times, namely to give up Fock space. As examples in spacetime dimensions 2 and 3 show, the interaction picture is perfectly well-defined if the Hilbert space is adapted to the field theory rather than being taken a priori as Fock space.

With all this in mind just out of curiosity is Relativistic QED at finite times nonrenormalizable or we simply dont know how to apply renormalization.

We know for hydrodynamic and kinetic studies, where CTP works perturbatively.

We still don't know it in general, since QED has not yet been constructed perturbatively. But neither is there any evidence that it would be impossible. See https://www.physicsoverflow.org/21328/

 

[physwiz222]

I am understanding the OP's question to refer to the fact that QFT texts rarely discuss what form finite time evolution might take in QFT.

In other words why we generally seem to avoid formulating a state $\rho(T)$ at some time $T$ in even a semi-explicit manner and solving for its evolution over some finite time interval: $U(t-T)\rho(T)$. This is quite unlike non-relativistic QM where this is often sketched, even though in many cases it could not be explicitly computed. Non-relativistic QM texts will cover cases outside of the overlap of asymptotic states.

I would give four initial reasons for this:

1. In QFT it is very difficult to characterise states that are not asymptotic. In non-relativistic QM of N-particles, pure states at any given time have the form $\psi(x_{1},\ldots,x_{N})$, i.e. some function in $L^{2}(\mathbb{R}^{3N})$.
   In QFT however we don't even know the Hilbert space for states in general and thus cannot even guess their rough functional form. At asymptotic times we know they are like free field states and thus can be represented as states in a Fock space of a finite number of particle species. At intermediate times it is an open problem of mathematical physics what form the states take, since we know that in general they cannot be understood in terms of particle states.
2. For asymptotic times we have the LSZ formalism that tells us we may compute the overlap between asymptotic states, i.e. the S-matrix, by computing Amputated n-point correlation functions. Since these n-point functions are expressions involving only finite products of local fields, we have further theorems showing they may be computed via Feynman graph techniques perturbatively.
   We have no such calculational recipe in general for finite time evolution.
3. Finite time evolution requires further renormalisations beyond those of asymptotic S-matrix calculations. A finite time propogator $U(t_{2},t_{1})$ requires formulating the theory on constant time slices, which due to the highly singular nature of the fields introduces new singularities. The finite time Green's functions that replace the usual propogator don't have the nice properties that make all-order BPHZ style renormalisation possible. Thus we don't have a general renormalisation prescription.
4. For QED in particular we have to deal with superselection sectors caused by an infrared cloud of photons surrounding a given electron. This means that nonperturbatively the relevant states are not Lorentz irreps and thus we lose all the simplifications such a simple representation theory brings.
   In fact non-perturbatively the evolution is most likely some non-unitary contractive Markovian evolution* about which very little is currently know.
   
   *One may understand this evolution in simpler terms as one involving a "perfect/exact" form of decoherence.

This is just a brief introduction to the issues involved. It's hard to explain in one go, let me know if you have questions.

What about Closed Time Path can that be used tor Scattering at Intermediate Times. Also what about just using the time evolution operator to evolve states Without taking the infinite time limit. Why cant these be used.

 

[physwiz222]

It's true that there are many mathematical problems unsolved in QFT. However, one must also say that in the vacuum QFT the calculation of the time-evolution (the initial-value problem) is pretty useless since there are no observables it depicts.

Things change, when it comes to the question of many-body physics. In relativistic heavy-ion collisions in fact we have to calculate the time evolution to a certain extent. For realistic simulations of the hot and dense fireball the most "fundamental" thing that's feasible computationally is the solution of quantum Boltzmann-Uehling Uhlenbeck equations on both the partonic and the hadronic level. There has a lot of progress over the last few decades. Among other things multi-particle collision terms (beyond the 2->2 level) can now be simulated accurately. Newest developments are mostly centered around "spin transport" and the derivation of hydrodynamics from relativistic kinetic theory. Also there there was tremendous progress over the last few years. Among other things now there's also a stable causal formulation of first-order ("Navier-Stokes") relativistic hydrodynamics as well as systematic approximations of higher orders. Of course also on the hydro level there's a lot of interest in spin and magneto-hydrodynamics.

On a more fundamental level one can solve toy models, using the real-time formalism of many-body off-equilibrium QFT, leading to Kadanoff-Baym equations (from which the above mentioned transport equations can be derived through gradient expansion of the Wigner representation of the contour Green's functions).

Two recent books about these topics are

G. S. Denicol and D. H. Rischke, Microscopic Foundationsof
Relativistic Fluid Dynamics, Springer, Cham (2021),
https://doi.org/10.1007/978-3-030-82077-0

W. Cassing, Transport Theories for Strongly-Interacting
Systems: Applications to Heavy-Ion Collisions, Springer,
Cham (2021),
https://doi.org/10.1007/978-3-030-80295-0

For the Interacting Vacuum wouldn’t the important observables be the Particle Number Density and Energy Density as functions of position and time.

 

[A. Neumaier]

For the Interacting Vacuum wouldn’t the important observables be the Particle Number Density and Energy Density as functions of position and time.

In a vacuum state there is nothing to be observed. For the interacting vacuum the particle number density is zero.

 

[physwiz222]

In a vacuum state there is nothing to be observed. For the interacting vacuum the particle number density is zero.

How do we know this. This interacting vacuum Ω is nothing like the free vacuum and is a complicated object from what I hear so how do you know the Particle Density n(r,t) or the Energy density is 0.