# What is the basic scheme of quantum field theories?

• I
Gold Member
Can anybody tell me a similar basic scheme for quantum field theory? For example, in QED, Is there a basic differential equation? What about QCD?

I'll add yet another perspective, since you asked about formulating QFT in terms of differential equations. As mentioned by @DarMM in multiple posts here, the main observables of a QFT are the correlation functions, which are related to experimental quantities. It turns out that the correlation functions of a quantum field theory satisfy a (generally infinite) set of differential equations called the Schwinger-Dyson equations. In principle, one could consider the solution to these equations as giving you the same information as any other method of calculating correlation functions. Whether this is actually the easiest approach depends on the problem, but sometimes numerically solving a truncated subset of them or solving them in some approximation scheme can be useful.

jonjacson
Gold Member
Well, they don't have a unique quantum version. But they have a parameterized family of quantum versions with an infinite number of parameters, most of them being irrelevant except at super high energies. We had discussed this in another thread; see https://www.physicsforums.com/threa...um-field-theory-comments.925220/#post-5844345 and the discussion following that post.
I know this and it is true, but I think this just depends on what you call a "quantum version", i.e. for ##\phi^{6}## in 4D there simply isn't a self-adjoint Hamiltonian whose highest power of the scalar field is the sixth power and thus the quantum version/analogue of that classical theory doesn't exist.

That one can order by order enlarge the space of terms to include ##\phi^{8}, \phi^{10}, \cdots## while keeping the theory finite perturbatively is true, but it isn't what this sort of nonexistence refers to. It refers to there simply being no quantum theory with a Hamiltonian of that form.

dextercioby
It refers to there simply being no quantum theory with a Hamiltonian of that form.
Well, renormalization destroys this form anyway. Your example of Gross-Neveu shows that ''form'' is not something sensible to discuss.

But you have to ask yourself - why is renormalization so fundamental that only such theories are valid?
Renormalizable theories are simply distinguished by having a natural parameterization with few parameters.

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For example, in QED, Is there a basic differential equation?
What remains of these after renormalization runs under the name of operator product expansions.

dextercioby and jonjacson
From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

"The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory." G. Scharf, Finite quantum electrodynamics. The causal approach, 1995.
You may think this is the case.

But on the level of rigor customary in theoretical physics, quantum field dynamics at finite time is actually well-defined in terms of the so-called closed time path (CTP) approach; see, e.g.,

E. Calzetta and B.L. Hu, Nonequilibrium quantum field theory, Cambridge Univ. Press, New York 2008.

dextercioby
Gold Member
Well, renormalization destroys this form anyway.
Does it? For example in Glimm's construction of ##\phi^{4}_{3}## the Hamiltonian on the interacting Hilbert space still has the same form. What is super-renormalization on Fock space turns out to be simple Wick ordering on the interacting Hilbert space.

meopemuk
But on the level of rigor customary in theoretical physics, quantum field dynamics at finite time is actually well-defined in terms of the so-called closed time path (CTP) approach; see, e.g.,

E. Calzetta and B.L. Hu, Nonequilibrium quantum field theory, Cambridge Univ. Press, New York 2008.

I briefly looked into this book and didn't find what I was looking for. I am interested in a QFT Hamiltonian that would generate the time evolution of simple interacting systems, e.g., two charges. So that using this Hamiltonian one can describe time-dependent wave functions of colliding particles in the interaction region in addition to the usual S-matrix. As far as I understand, standard QFT does not permit such a description. (Such a description is possible in the dressed particle approach to QFT; see vol. 3 of my book.)

It seems that the book you recommended is not interested in such simple few-particle systems. The authors are focused on applying QFT to nonequilibrium statistical physics and such complex phenomena as heavy ion collisions, early universe cosmology, Bose-Einstein condensation, dissipation, entropy, decoherence. All this is interesting and exciting, but I would like to stick to the old-fashioned approach: learn simple systems (mechanics of few particles) first before turning to complicated ones (nonequilibrium statistical physics and cosmology).

I couldn't understand how CPT approach can replace the good old Hamiltonian as a method for describing time evolution. Perhaps you can explain that? In my understanding, if we want a unitary and relativistically-invariant time evolution, then there is nothing better than a Hamiltonian satisfying the usual commutation relations with other Poincare generators. If you describe the time evolution in any other way, then you lose either unitarity or relativistic invariance or both.

Eugene.

Gold Member
That's true, its the old view of renormalisation. But you have to ask yourself - why is renormalization so fundamental that only such theories are valid? Wilsons view that its simply the low energy limit of a possibly non-renormalizeable theory makes more sense. Of course nature making sense to me may not be how nature works.
It's actually quite simple in this view. Renormalization is simply a procedure of correctly constructing the Hamiltonian or Path Integral measure. Only those theories for which this procedure works (i.e. are renormalizable) exist.

For example in ##d = 3## let's say the equation:
$$(\partial^{2} - m^{2})\phi = \frac{\lambda}{3!}\left[\phi^{3}\right]_R$$
has a solution, but
$$(\partial^{2} - m^{2})\phi = \frac{\lambda}{7!}\left[\phi^{7}\right]_R$$
doesn't. Perturbatively this nonexistence will show up as non-renormalizability, but "fundamentally" it's simply that the equations don't have solutions. Quantum Fields are so singular that the space of PDEs with solutions is smaller than for Classical Fields. And that's all there is to it in this view.

dextercioby and bhobba
bolbteppa
You may think this is the case.

But on the level of rigor customary in theoretical physics, quantum field dynamics at finite time is actually well-defined in terms of the so-called closed time path (CTP) approach; see, e.g.,

E. Calzetta and B.L. Hu, Nonequilibrium quantum field theory, Cambridge Univ. Press, New York 2008.

Trying to describe the time evolution of a relativistic quantum system contradicts the most fundamental principles of quantum mechanics + relativity:

"The momentum can figure in a consistent theory only for free particles; for these it is conserved, and can therefore be measured with any desired accuracy. This indicates that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta, polarisations) of free particles: the initial particles which come into interaction, and the final particles which result from the process"
Berestetskii Lifshitz Quantum Electrodynamics Section 1.

Section 1 is devoted to justifying this.

QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future. From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

"The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory." G. Scharf, Finite quantum electrodynamics. The causal approach, 1995.

Eugene.

As far as I understand, standard QFT does not permit such a description.
.

The book above justifies this in section 1 as referenced.

In classical mechanics I would say:
a particle can have any initial position and velocities, Newton laws give you the evolution of the particle:
F=ma , is the basic equation, if you know the forces acting on the particle by solving this equation you get the future values for velocity and position

In quantum mechanics:
The "particle" is characterized by a wavefunction that can be used to calculate the probability of finding the particle at certain position or momentum.
If you know the wavefunction you plug it into the Schrodinger equation and you get the evolution on time of this wavefunction.

Can anybody tell me a similar basic scheme for quantum field theory? For example, in QED, Is there a basic differential equation? What about QCD?

The basic scheme is still just the Schrodinger equation, for relativistic Hamiltonians, however relativity forces one to restrict the probability distribution interpretation of wave functions to momentum space for free particles and to work with wave functions for variable numbers of particles (section 1 cited above), for which second quantization is most convenient - and the Heisenberg picture for second quantization operators even more convenient, the reference above being a good start.

Gold Member
Quite a morass of constructions above. To go back to basics of "What is the basic scheme of quantum field theories?", the basics of a free field bosonic QFT can be presented by a generating function for time-ordered free quantum field as $$\langle 0|T[\mathrm{e}^{\mathrm{i}\lambda\hat\phi_f}]|0\rangle=\mathrm{e}^{-\lambda^2(f^*,f)_F/2},$$ where ##(f,g)_F## is the Feynman propagator smeared antilinearly by ##f## and linearly by ##g##, so that the n-th derivative w.r.t. ##\lambda## of this at ##\lambda=0## will give ##\langle 0|T[\hat\phi_f^{\,n}]|0\rangle##. I'll mention that ##(f,g)_F## is also proportional to Planck's constant.
An aside: the subscript ##f## in ##\hat\phi_f## is usually called a test function (take ##f## to be a smooth function that also has a smooth Fourier transform), which is an index or coordinate for a measurement. We can think loosely of the usual quantum field object ##\hat\phi(x)## as what we would get if we took ##f## to be a Dirac delta function at ##x##, but it's better to avoid doing that because a Dirac delta function is not a smooth function. We can think of ##f## as a modulation that is applied to a vacuum state or as like the window functions of signal analysis, depending on how we use it.​
The time-ordered form, however, erases information about the algebraic structure, which we can take a first step towards by presenting a generating function form without time-ordering, $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda\hat\phi_f}|0\rangle=\mathrm{e}^{-\lambda^2(f^*,f)/2}.$$ Everything about the free quantum field algebra can be fixed by a single equation that generalizes the single operator case, $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\phi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\phi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\phi_{f_n}}|0\rangle=\exp\left[-\sum_{i,j}\lambda_i\lambda_j(f_i^*,f_j)/2-\sum_{i<j}\left[(f^*_i,f_j)-(f^*_j,f_i)\right]/2\right],$$ where in the exponent on the r.h.s. the first term is what would be classically called noise and the second term is the measurement incompatibility that characterizes quantum theory.
For an interacting quantum field ##\hat\xi_f##, in principle all the Feynman diagrams, regularization, and renormalization are trying to do is to deform this expression to give us a new expression, $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\xi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\xi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\xi_{f_n}}|0\rangle=\,\cdots,$$ a function of all the ##\lambda_i##'s and ##f_i##'s, that we can use to generate any VEV (vacuum expectation Value) and which fixes the algebraic structure for ##\hat\xi_f##, just as we saw above for ##\hat\phi_f##. That gets complicated because for at least the last 70 years the Feynman integral formalism has insisted on using powers of the operator-valued distribution ##\hat\phi(x)##, which is frankly a mathematically stupid thing to do and requires all sorts of desperate first-aid to fix the resulting problems, instead of figuring out ways to construct deformations that use only the well-defined objects, the test functions ##f_i##. In principle the VEVs that are generated by $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\xi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\xi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\xi_{f_n}}|0\rangle$$ are commonly supposed to be measurable by experiment (unless someone insists that only the S-matrix can be measured, which describes how a state changes between times ##t=\pm\infty##, so go figure why anyone would think that), although they are not exactly close to the recorded measurements of a signal voltage on a wire that is attached to some kind of fancy material coupled to a region of space-time deep within an experimental apparatus.
I'm not sure whether that will be helpful for you or for anyone else, but I find it helpful for me. I particularly find it helpful to think of QFT, free or interacting, as a signal analysis formalism, because all our experimental raw data comes into a computer as voltages on signal lines, which are converted to a binary representation and stored.
All the calculations above are essentially easy to check by using a Baker-Campbell-Haussdorf identity, setting ##\hat\phi_f=a_{f^*}+a_f^\dagger## in terms of creation and annihilation operators and using the commutation relation ##[a_f,a_g^\dagger]=(f,g)##.

jonjacson
Gold Member
Trying to describe the time evolution of a relativistic quantum system contradicts the most fundamental principles of quantum mechanics + relativity:

Section 1 is devoted to justifying this.

The book above justifies this in section 1 as referenced.

The basic scheme is still just the Schrodinger equation, for relativistic Hamiltonians, however relativity forces one to restrict the probability distribution interpretation of wave functions to momentum space for free particles and to work with wave functions for variable numbers of particles (section 1 cited above), for which second quantization is most convenient - and the Heisenberg picture for second quantization operators even more convenient, the reference above being a good start.

It seems to me that these considerations only make sense if one either (1) defines "particles" in a strict sense which is not useful in an interacting QFT (which generically does not have particles in the sense often meant), or (2) demands that the theory is "relativistic" in the sense that one does not have a UV cutoff, in which case I do not know of any realistic interacting relativistic QFT in (3+1) dimensions.

If I define a quantum field theory with the appropriate regulators, I can describe time-dependent phenomena (equilibrium or non-equilibrium) just fine. You may say that I have broken Lorentz invariance, which is true (after all Lorentz invariant theories have not been proven to exist), but for suitably large cutoffs all low-energy phenomena exhibit emergent Lorentz invariance anyways. And these calculations agree with experiment (see the many thousands of papers using time-dependent methods in QFT published every year).

(I also don't trust a QFT textbook/lecture which does not cover the renormalization group. This field has evolved in the past 50 years!)

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meopemuk
And these calculations agree with experiment (see the many thousands of papers using time-dependent methods in QFT published every year).

Perhaps you can mention a couple of papers where a full renormalized relativistic QFT (i.e., whose Hamiltonian has cutoff-dependent divergent counterterms) was used to calculate time-dependent processes, and the results agree with experiment?

Thanks.
Eugene.

jonjacson
I'll add yet another perspective, since you asked about formulating QFT in terms of differential equations. As mentioned by @DarMM in multiple posts here, the main observables of a QFT are the correlation functions, which are related to experimental quantities. It turns out that the correlation functions of a quantum field theory satisfy a (generally infinite) set of differential equations called the Schwinger-Dyson equations. In principle, one could consider the solution to these equations as giving you the same information as any other method of calculating correlation functions. Whether this is actually the easiest approach depends on the problem, but sometimes numerically solving a truncated subset of them or solving them in some approximation scheme can be useful.

So interesting, thanks. An infinite set of differential equations looks like something not easy to solve definitely :(

What remains of these after renormalization runs under the name of operator product expansions.
Thanks.

Quite a morass of constructions above. To go back to basics of "What is the basic scheme of quantum field theories?", the basics of a free field bosonic QFT can be presented by a generating function for time-ordered free quantum field as $$\langle 0|T[\mathrm{e}^{\mathrm{i}\lambda\hat\phi_f}]|0\rangle=\mathrm{e}^{-\lambda^2(f^*,f)_F/2},$$ where ##(f,g)_F## is the Feynman propagator smeared antilinearly by ##f## and linearly by ##g##, so that the n-th derivative w.r.t. ##\lambda## of this at ##\lambda=0## will give ##\langle 0|T[\hat\phi_f^{\,n}]|0\rangle##. I'll mention that ##(f,g)_F## is also proportional to Planck's constant.
An aside: the subscript ##f## in ##\hat\phi_f## is usually called a test function (take ##f## to be a smooth function that also has a smooth Fourier transform), which is an index or coordinate for a measurement. We can think loosely of the usual quantum field object ##\hat\phi(x)## as what we would get if we took ##f## to be a Dirac delta function at ##x##, but it's better to avoid doing that because a Dirac delta function is not a smooth function. We can think of ##f## as a modulation that is applied to a vacuum state or as like the window functions of signal analysis, depending on how we use it.​
The time-ordered form, however, erases information about the algebraic structure, which we can take a first step towards by presenting a generating function form without time-ordering, $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda\hat\phi_f}|0\rangle=\mathrm{e}^{-\lambda^2(f^*,f)/2}.$$ Everything about the free quantum field algebra can be fixed by a single equation that generalizes the single operator case, $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\phi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\phi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\phi_{f_n}}|0\rangle=\exp\left[-\sum_{i,j}\lambda_i\lambda_j(f_i^*,f_j)/2-\sum_{i<j}\left[(f^*_i,f_j)-(f^*_j,f_i)\right]/2\right],$$ where in the exponent on the r.h.s. the first term is what would be classically called noise and the second term is the measurement incompatibility that characterizes quantum theory.
For an interacting quantum field ##\hat\xi_f##, in principle all the Feynman diagrams, regularization, and renormalization are trying to do is to deform this expression to give us a new expression, $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\xi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\xi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\xi_{f_n}}|0\rangle=\,\cdots,$$ a function of all the ##\lambda_i##'s and ##f_i##'s, that we can use to generate any VEV (vacuum expectation Value) and which fixes the algebraic structure for ##\hat\xi_f##, just as we saw above for ##\hat\phi_f##. That gets complicated because for at least the last 70 years the Feynman integral formalism has insisted on using powers of the operator-valued distribution ##\hat\phi(x)##, which is frankly a mathematically stupid thing to do and requires all sorts of desperate first-aid to fix the resulting problems, instead of figuring out ways to construct deformations that use only the well-defined objects, the test functions ##f_i##. In principle the VEVs that are generated by $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\xi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\xi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\xi_{f_n}}|0\rangle$$ are commonly supposed to be measurable by experiment (unless someone insists that only the S-matrix can be measured, which describes how a state changes between times ##t=\pm\infty##, so go figure why anyone would think that), although they are not exactly close to the recorded measurements of a signal voltage on a wire that is attached to some kind of fancy material coupled to a region of space-time deep within an experimental apparatus.
I'm not sure whether that will be helpful for you or for anyone else, but I find it helpful for me. I particularly find it helpful to think of QFT, free or interacting, as a signal analysis formalism, because all our experimental raw data comes into a computer as voltages on signal lines, which are converted to a binary representation and stored.
All the calculations above are essentially easy to check by using a Baker-Campbell-Haussdorf identity, setting ##\hat\phi_f=a_{f^*}+a_f^\dagger## in terms of creation and annihilation operators and using the commutation relation ##[a_f,a_g^\dagger]=(f,g)##.

Thanks.

Why do you think it hasn't changed in 70 years?

Gold Member
Why do you think it hasn't changed in 70 years?
I would say because it clearly works. We know rigorously that defining products of operator valued distributions like ##\phi^{4}## is tricky, but it is well-defined and the standard methods of doing things have been proven to be valid in any rigorous construction we've achieved.

So we know the "textbook" way of doing things is quite good.

Doing things rigorously/correctly is extremely difficult and abstract and hard to get numbers out of and even when you do it they're pretty much the same numbers. The well-defined objects that you should be using, local operator algebras, are very abstract and not easy to work with.

Gold Member
Perhaps you can mention a couple of papers where a full renormalized relativistic QFT (i.e., whose Hamiltonian has cutoff-dependent divergent counterterms) was used to calculate time-dependent processes, and the results agree with experiment?

Thanks.
Eugene.

What issues do you have with the Calzetta & Hu book cited by Arnold Neumaier? It has about 1000 references in it, and the book contains many applications to relativistic heavy ions collisions, cosmology, chiral condensates, and condensed matter (the latter often has a relativistic low-energy description). I just searched through my copy and found many examples where agreement with experiments was cited.

I might be exaggerating with "thousands," but calculating dynamics in a QFT seems very standard to me. It might come down to what you consider a time-dependent calculation. Does linear response count? I suppose you wouldn't count calculations of relaxation rates or decay times? Would the measurement of a spectral function (basically the Fourier transform of a two-point function) count? I'd consider all of these to be dynamical info about time-evolution of the theory, but perhaps it is simply not what you mean with your statement.

Also, although you asked about theories with "divergent counterterms" (implying perturbation theory), there are exactly solvable interacting relativistic QFTs where we can specify the dynamics exactly. The only experimental applications I know of are condensed matter systems (probed at long wavelengths), but my point is that the question is well-posed.

dextercioby and protonsarecool
Gold Member
Why do you think it hasn't changed in 70 years?
1949 is a slightly arbitrary date, because people knew there were infinities almost immediately after the idea of quantizing interacting field theories got off the ground in the late 1920's, but that's when something like renormalization first became a "solution", and we've basically not got away from that way of working with interacting QFTs. One can find work on different kinds of generalized functions, and other attempts to do things right, but they still don't quite work.

You can look at my post https://quantumclassical.blogspot.com/2019/02/lost-in-math-review-in-perspective.html, of a few days ago, where you can find the following, towards the end:
How, then, are we to avoid those infinities that have to be regularized and renormalized? We want a way to construct generating functions for an idealized state that gives us values for ##\rho(F_aF_bF_cF_dF_eF_f\cdots)## that are worth having for practical engineering. Since the 1950s, the starting point for this kind of thinking has been the Wightman axioms, which we can present, adapted from Rudolph Haag's book Local Quantum Physics, as:
• A Hilbert space H supports a unitary representation of the Poincar é group; there is a unique Poincar é invariant vacuum state, of lowest energy.
• Quantum fields are operator-valued distributions, linear maps from a measurement description space (the a, b, c, ...) into operators ##F_a, F_b, F_c, ...## in a ##*##-algebra A.
• Quantum fields support a nontrivial representation of the Poincaré group.
• Microcausality: commutativity at space-like separation (no faster-than-light signalling).
• Completeness: the action of the quantum field is irreducible (that is, states must be pure).
• Time-slice axiom (the state now determines the future state).
The mentions of the Poincaré group and of microcausality are empirically quite well justified, but at least three of these constraints are blatantly a priori, introduced more to make the math work nicely than to make the math physically useful: (1) that the vacuum state should be of lowest energy (thermal equilibrium does not satisfy this axiom); (2) that the ##F_a, F_b, F_c, ...## must be linear functionals of the a, b, c, ... (classically, ##\rho(F_aF_b)## can be understood to be an ##F_a## response to an ##F_b## modulation, which would not be expected to be linear in both a and b); and (3) completeness (again, a thermal equilibrium does not satisfy this axiom, but also, if any degrees of freedom are traced out, which dark matter and dark energy are, the resulting state is a mixed state: we can, for example, usefully measure just the electromagnetic field, only inferring some aspects of, but not explicitly measuring, electric currents). One consequence of removing the second, linearity, can be found in arXiv:1507.08299, though I think I might construct this paper in a somewhat different way now than I did four years ago. Others of these axioms could also be weakened or changed, perhaps even one or more might have to be strengthened to make the resulting system a better engineering tool: whatever must be done to allow us to match the experimental values must be done. The three changes above already give us a plethora of models to characterize and to check off against nature.​
Needless to say, feel free to ignore this. I typed out my comment above, and this comment, as much to see for myself how it would come out.

jonjacson
jonjacson
1949 is a slightly arbitrary date, because people knew there were infinities almost immediately after the idea of quantizing interacting field theories got off the ground in the late 1920's, but that's when something like renormalization first became a "solution", and we've basically not got away from that way of working with interacting QFTs. One can find work on different kinds of generalized functions, and other attempts to do things right, but they still don't quite work.

You can look at my post https://quantumclassical.blogspot.com/2019/02/lost-in-math-review-in-perspective.html, of a few days ago, where you can find the following, towards the end:
How, then, are we to avoid those infinities that have to be regularized and renormalized? We want a way to construct generating functions for an idealized state that gives us values for ##\rho(F_aF_bF_cF_dF_eF_f\cdots)## that are worth having for practical engineering. Since the 1950s, the starting point for this kind of thinking has been the Wightman axioms, which we can present, adapted from Rudolph Haag's book Local Quantum Physics, as:
• A Hilbert space H supports a unitary representation of the Poincar é group; there is a unique Poincar é invariant vacuum state, of lowest energy.
• Quantum fields are operator-valued distributions, linear maps from a measurement description space (the a, b, c, ...) into operators ##F_a, F_b, F_c, ...## in a ##*##-algebra A.
• Quantum fields support a nontrivial representation of the Poincaré group.
• Microcausality: commutativity at space-like separation (no faster-than-light signalling).
• Completeness: the action of the quantum field is irreducible (that is, states must be pure).
• Time-slice axiom (the state now determines the future state).
The mentions of the Poincaré group and of microcausality are empirically quite well justified, but at least three of these constraints are blatantly a priori, introduced more to make the math work nicely than to make the math physically useful: (1) that the vacuum state should be of lowest energy (thermal equilibrium does not satisfy this axiom); (2) that the ##F_a, F_b, F_c, ...## must be linear functionals of the a, b, c, ... (classically, ##\rho(F_aF_b)## can be understood to be an ##F_a## response to an ##F_b## modulation, which would not be expected to be linear in both a and b); and (3) completeness (again, a thermal equilibrium does not satisfy this axiom, but also, if any degrees of freedom are traced out, which dark matter and dark energy are, the resulting state is a mixed state: we can, for example, usefully measure just the electromagnetic field, only inferring some aspects of, but not explicitly measuring, electric currents). One consequence of removing the second, linearity, can be found in arXiv:1507.08299, though I think I might construct this paper in a somewhat different way now than I did four years ago. Others of these axioms could also be weakened or changed, perhaps even one or more might have to be strengthened to make the resulting system a better engineering tool: whatever must be done to allow us to match the experimental values must be done. The three changes above already give us a plethora of models to characterize and to check off against nature.​
Needless to say, feel free to ignore this. I typed out my comment above, and this comment, as much to see for myself how it would come out.

So even if it is said QFT is a tremendous success, and it is very precise, it is still an unfinished theory with some problems that must be adressed, Is that correct?

Peter Morgan
Gold Member
@Peter Morgan , the basic scheme of QFT is not your personal ideas about how it should be done, especially not in a thread where somebody is asking for a basic summary of the field. I think this goes even more so for you @meopemuk .

The modifications you mention have been well investigated. For example not requiring the fields to be distributions, but possibly hyperfunctions, ultradistributions or other such generalised functions was already well looked into by Glimm and Jaffe in the 1970s. The Wightman axioms are supposed to be for the vacuum sector, of course they don't take into account Thermal states. For that one can turn to the Haag-Kastler axioms.

So even if it is said QFT is a tremendous success, and it is very precise, it is still an unfinished theory with some problems that must be adressed, Is that correct?
Yes, although many theories in physics are like this. The primary practical problem with QFT is a better understanding of nonperturbative effects, i.e. how theories work outside of Taylor expansions and a better understanding of calculating processes that aren't scattering experiments. Both of these are quite developed but there's a lot of exciting work still to be done.

dextercioby
jonjacson
@Peter Morgan , the basic scheme of QFT is not your personal ideas about how it should be done, especially not in a thread where somebody is asking for a basic summary of the field. I think this goes even more so for you @meopemuk .

The modifications you mention have been well investigated. For example not requiring the fields to be distributions, but possibly hyperfunctions, ultradistributions or other such generalised functions was already well looked into by Glimm and Jaffe in the 1970s. The Wightman axioms are supposed to be for the vacuum sector, of course they don't take into account Thermal states. For that one can turn to the Haag-Kastler axioms.

Yes, although many theories in physics are like this. The primary practical problem with QFT is a better understanding of nonperturbative effects, i.e. how theories work outside of Taylor expansions and a better understanding of calculating processes that aren't scattering experiments. Both of these are quite developed but there's a lot of exciting work still to be done.

Do you have a link, or can you list the most important questions that should be clarified or solved?

Gold Member
Do you have a link, or can you list the most important questions that should be clarified or solved?
As @DarMM says, I've strayed into my own personal territory. My first post was all elementary calculations, although even by doing one set of calculations instead of another one can put quite a slant on things. He's also right that hyperfunctions, et cetera, were tried and more-or-less failed, or at least failed to catch on, in the 1970s, so some new variant of that would be required, but such an approach might still work.
I'm just a one man show, so I move very slowly. The arXiv papers mentioned in the blog post I linked to above are the current "literature" on my PoV, so by the rules of physicsforums they're not admissible because they're unpublished (the 1709.06711 might make it into Physica Scripta, if the referee is OK with the changes I'm putting into accommodate their comments, but not yet!), which is fair enough but it's one reason why I don't come here very often. I liked your question and I liked the way you responded to the answers, so I jumped in even though it was bound to come to this if anyone led me on! I can appeal to Dirac as a determined opponent of renormalization, despite @DarMM's entirely correct comments above, but of course that was when he was old, that aspect of his thinking was largely ignored, and, as far as I know, he never had anything constructive to say about how to do things differently, at least not that worked.
If you'd like to find me on Facebook, or ask in comments on the blog post I linked to, by all means do, but no worries.

DarMM and Greg Bernhardt
Gold Member
Don't worry @Peter Morgan , I've gone overboard in B and I threads myself (In the form: What's an operator you ask, well consider endomorphisms, blah, blah), it's easy to do so when you're enthusiastic.

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dextercioby and Peter Morgan
Gold Member
@DarMM, regarding the Haag-Kastler axioms, what you find depends quite a lot on where you look. Section III.1 of Haag's Local Quantum Physics, for example, relates the Haag-Kastler axioms to the corresponding Wightman axioms, saying that "we shall keep the assumptions pertaining to [the Poincaré group], most importantly the existence of a vacuum state and the positivity of the energy." Later in the book he discusses thermal states in terms of analyticity, but I find it very helpful to present a thermal state at a temperature ##kT## as I did above for the vacuum state, which, to recall, was $$\langle 0|\mathrm{e}^{\mathrm{i}\lambda_1\hat\phi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\phi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\phi_{f_n}}|0\rangle=\exp\left[-\sum_{i,j}\lambda_i\lambda_j(f_i^*,f_j)/2-\sum_{i<j}\left[(f^*_i,f_j)-(f^*_j,f_i)\right]/2\right],$$ whereas for a thermal state we have exactly the same structure except that the noise at temperature ##T## is increased from ##(f^*,f)## to ##(f^*,f)_{kT}##, $$\rho_{kT}(\mathrm{e}^{\mathrm{i}\lambda_1\hat\phi_{f_1}}\mathrm{e}^{\mathrm{i}\lambda_2\hat\phi_{f_2}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat\phi_{f_n}})=\exp\left[-\sum_{i,j}\lambda_i\lambda_j(f_i^*,f_j)_{kT}/2-\sum_{i<j}\left[(f^*_i,f_j)-(f^*_j,f_i)\right]/2\right],$$ with measurement incompatibility unchanged. All straightforward computation, which can be found (in a slightly less refined form) in my "A succinct presentation of the quantized Klein–Gordon field, and a similar quantum presentation of the classical Klein–Gordon random field", Physics Letters A 338 (2005) 8–12, https://arxiv.org/abs/quant-ph/0411156, with the pre-inner product for the Klein-Gordon field at temperature ##T## being $$(f,g)_{kT}=\hbar\int\tilde f^*(k)\coth\left(\frac{\hbar k_0}{2kT}\right)\tilde g(k)2\pi\delta(k{\cdot}k{-}m^2)\theta(k_0)\frac{\mathrm{d}^4k}{(2\pi)^4}.$$ Just take out the ##\coth(\cdot)## factor for the vacuum state pre-inner product. It's not specially hard to show that this thermal state is infinite energy.
To be clear, I'm not at all saying you're wrong, I'm only pointing out that exactly what one takes the Wightman axioms to be makes a significant difference, because the state just constructed is not only infinite energy, it's also a mixed state. If one relinquishes both those axioms, we suddenly have a whole slew of states that satisfy the Wightman-(PositiveEnergy, Irreducibility) axioms, not just thermal states, including many that are Poincaré invariant, and many that are not Gaussian at all, No, No, No. Eliminating all of them as not useful can be lots of fun for everyone.

@jonjacson: Please edit quotes from long answers and keep only the small part of the quote needed to address what you want to comment. Quoting long passages without reason pollutes the browser, a nuisance especially when using a smartphone.

Trying to describe the time evolution of a relativistic quantum system contradicts the most fundamental principles of quantum mechanics + relativity:
But not the principles of relativistic quantum field theory. Given a realization of the Wightman axioms, time evolution is given by shifting all field arguments in their time argument by the same amount.

there are exactly solvable interacting relativistic QFTs where we can specify the dynamics exactly.
Only in 2-dimensional spacetime.

the Calzetta & Hu book cited by Arnold Neumaier? It has about 1000 references in it, and the book contains many applications to relativistic heavy ions collisions, cosmology, chiral condensates, and condensed matter (the latter often has a relativistic low-energy description). I just searched through my copy and found many examples where agreement with experiments was cited.

I might be exaggerating with "thousands," but calculating dynamics in a QFT seems very standard to me. It might come down to what you consider a time-dependent calculation. Does linear response count? I suppose you wouldn't count calculations of relaxation rates or decay times? Would the measurement of a spectral function (basically the Fourier transform of a two-point function) count? I'd consider all of these to be dynamical info about time-evolution of the theory

Yes, indeed.

I briefly looked into this book and didn't find what I was looking for. I am interested in a QFT Hamiltonian that would generate the time evolution of simple interacting systems, e.g., two charges.
Interacting QFT has no notion of particles (except asymptotically at times ##\pm\infty##. Thus it also has no concept of two moving charges, only of states with charge number 2. This is underlying the fact that the foundations of particle physics are expressed in terms of quantum field theory, and not of quantum particle theory. Particles can be used only in an approximation where their dynamics can be considered to be essentially free most of the time - i.e., in scattering experiments, or when the mean free paths of the particles (and in practice, of appropriately defined quasiparticles not figuring in the underlying field description) are long enough to allow for the use of the asymptotic picture.
So that using this Hamiltonian one can describe time-dependent wave functions of colliding particles in the interaction region in addition to the usual S-matrix.
The time-dependent dynamics of interacting relativistic particles is traditionally described by means of Kadanoff-Baym theory - Hendrik van Hees (@vanhees71) does this in his daily research -, based on the CTP description given in the book by Calzetta and Hu.
As far as I understand, standard QFT does not permit such a description.
This is because you want to understand it in terms of a Hamiltonian expressed in a Fock space. This is indeed impossible due to Haag's theorem. But as an operator on the renormalized Hilbert space (which by Haag's theorem is not a Fock space, though often constructible as a kind of limit of Fock spaces), the Hamiltonian is perturbatively well-defined. This is the case since the Wightman correlation functions are perturbatively well-defined, and the Hamiltonian can be specified in terms of the latter (as my comment on bolteppa's answer shows).
(Such a description is possible in the dressed particle approach to QFT; see vol. 3 of my book.)
Only in perturbation theory (which works for electromagnetic interactions but not for QCD), and only at the expense of compromising with the requirements of relativity theory. This is far from satisfactory, except apparently for you.

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dextercioby and DarMM
jonjacson
I can't find an edit button. I can see only Report, Quote and Reply to my own posts.

I can't find an edit button. I can see only Report, Quote and Reply to my own posts.
You can also edit your old contributions by clicking there on the edit button. Please remove the superfluous passages.

jonjacson
jonjacson
Can you show me where is the edit button? I feel so stupid at this moment.

Can you show me where is the edit button?
there are 4 buttons: edit - delete -report -bookmark
(Below other's entries, there are only the last two of these.)

Yes, to edit your old posts, click on edit.

If you want to reply to several parts of an answer, delete first the unneeded stuff, then copy and duplicate the whole quote environment, then edit out the parts not commented.

jonjacson
Well, I can't see it.

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Staff Emeritus
QFT does not allow you to calculate the time evolution, because the Hamiltonian (=the generator of time evolution) of QFT is badly damaged by the presence of divergent counterterms. QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future. From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

Lattice quantum field theory can in principle calculate time evolution. There might be problems in taking the continuum limit, though, so this calculation might not tell us anything about real quantum fields.

Lattice quantum field theory can in principle calculate time evolution. There might be problems in taking the continuum limit, though, so this calculation might not tell us anything about real quantum fields.

Textbook QED has no sensible relation to lattice QED. At large lattice spacing it is way too inaccurate, and at small lattice spacing, at a cell size still amenable to simulation, it shows already all numerical signs of triviality, i.e., it converges to a free theory rather than to QED. Thus there is no domain where lattice QED approximates QED in any meaningful way.

Eugene's book is about a particle approach to QED, so your comment is irrelevant for his purposes.

Well, I can't see it.
oh, this is different from what I see. Then currently you cannot edit old posts. Maybe editing is allowed only for science advisors? @Greg Bernhardt

oh, this is different from what I see. Then currently you cannot edit old posts. Maybe editing is allowed only for science advisors? @Greg Bernhardt

SA's have no time limit whereas regular membership has 60min.

Gold Member
Textbook QED has no sensible relation to lattice QED. At large lattice spacing it is way too inaccurate, and at small lattice spacing, at a cell size still amenable to simulation, it shows already all numerical signs of triviality, i.e., it converges to a free theory rather than to QED. Thus there is no domain where lattice QED approximates QED in any meaningful way.

Do you have a reference for this?

bolbteppa
Apologies for the nested quotes, for context:
... QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future... "The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. ... Essentially only scattering experiments are possible,

You may think this is the case.

But on the level of rigor customary in theoretical physics, quantum field dynamics at finite time is actually well-defined in terms of the so-called closed time path (CTP) approach
Trying to describe the time evolution of a relativistic quantum system contradicts the most fundamental principles of quantum mechanics + relativity:
"The momentum can figure in a consistent theory only for free particles; for these it is conserved, and can therefore be measured with any desired accuracy. This indicates that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta, polarisations) of free particles: the initial particles which come into interaction, and the final particles which result from the process"
But not the principles of relativistic quantum field theory. Given a realization of the Wightman axioms, time evolution is given by shifting all field arguments in their time argument by the same amount.

I was referring to the principles of relativistic quantum field theory - it is one thing for the fields to shift in their field arguments in time, which of course has to happen, it is another to actually describe (i.e. measure) what is going on at every time in the course of that time evolution in interacting theories, "quantum field dynamics at finite time" for anything not equivalent to scatting between free particle states, which amounts to free particles scattering "from the infinite past to the infinite future", is "just as unreal as the classical paths are in non-relativistic quantum mechanics", contradicting this is not permissible in the absence of a suitable level of rigor.

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bolbteppa
It seems to me that these considerations only make sense if one either (1) defines "particles" in a strict sense which is not useful in an interacting QFT (which generically does not have particles in the sense often meant), or (2) demands that the theory is "relativistic" in the sense that one does not have a UV cutoff, in which case I do not know of any realistic interacting relativistic QFT in (3+1) dimensions.

If I define a quantum field theory with the appropriate regulators, I can describe time-dependent phenomena (equilibrium or non-equilibrium) just fine. You may say that I have broken Lorentz invariance, which is true (after all Lorentz invariant theories have not been proven to exist), but for suitably large cutoffs all low-energy phenomena exhibit emergent Lorentz invariance anyways. And these calculations agree with experiment (see the many thousands of papers using time-dependent methods in QFT published every year).

(I also don't trust a QFT textbook/lecture which does not cover the renormalization group. This field has evolved in the past 50 years!)

If we can only measure things like free particle momenta, with anything else "just as unreal as the classical paths are in non-relativistic quantum mechanics", why should we expect the notion of particles in interacting qft to be useful if all that matters, according to the fundamental principles of QM and relativity, is the outcome (i.e. particles scattering off into free states) of such processes?

Without Lorentz/Poincare invariance, we have nothing - apart from, at most, non-relativistic QFT, i.e. second quantized non-relativistic quantum mechanics - the idea that "Lorentz invariant theories have not been proven to exist" is a realistic justification for ignoring one of the most important/successful principles in science is as realistic as saying 'no theory in science has been proven to exist'.

Concepts like cutoffs and regulators are related to the necessity of renormalization which is also a huge problem in classical ('relativistic') electromagnetism where, again, the issue is the point-particle model of physics, something relativity simply makes mandatory - again, without relativity we have nothing, especially dependent concepts like cutoffs (which also arise in 'relativistic' classical electromagnetism), and with relativity we unavoidably have a point particle model (and so renormalization issues, which is why it's not surprising string theories with their non-point-particle model are the only known way to deal with QM-GR renormalization issues).

That book does mention the renormalization group (of Gell-Mann and Low) when deriving the Landau pole issue in the second last chapter, advances in renormalization are pretty much not going to do anything but better bypass the more fundamental fact that infinities come from the point-particle nature of our models of (most) fundamental physics, which are inherently linked to relativity - a point which is undeniable in modern mainstream physics, and something modern research (strings etc) tries to deal with.