Time evolution in quantum field theories

[A. Neumaier]

unitary time evolution in quantum mechanics requires a well-defined Hermitian Hamiltonian.

It requires more than that: a well-defined, selfadjoint Hamiltonian. See http://arxiv.org/pdf/quant-ph/9907069 for a gentle introduction and counterexamples. An in depth discussion is given in Vol. 1 of the math physics treatise by Reed and Simon, or Vol.3 of the math physics treatise by Thirring.

I don't understand how CTP can do time evolution without a Hamiltonian. I would appreciate very much if you can explain that to me in simple terms.

It doesn't do without a Hamiltonian - only without a Hamiltonian expressed explicitly in terms of creation and annililation operators. Explaining this takes some preparations:

In an introductory textbook treatment of relativistic quantum field theory (QFT) one only finds a discussion of how to compute scattering information, since this is the most elementary and important output for studying the subnuclear world. Since scattering only concerns how things in the infinite past turn into things in the infinite future, it looks as if QFT had nothing to say about the finite-time evolution of quantum fields. But this is an illusion.

Quantum field theory is the theory of computing and interpreting expectations of products of the basic fields at different space-time arguments. In traditional renormalized perturbation theory, this is done by means of representing these expectations as infinite series of terms representable as weighted sums over multi-momentum integrals, commonly expressed through various Feynman diagrams.

The terms in this series are given a well-defined mathematical sense by a renormalization process consisting in a carefully taken limit of the appropriate sums of integrals. This looks untidy in most textbook treatments of renormalization (for a careful treatment, see the books by Salmhofer or by Scharf), where there is talk about subtracting infinities, which gives the whole procedure an air of arbitrariness. But renormalization is nothing but a more complex version of the elementary stuff we learn early in our science education about how to subtract infinities arising when setting x=1 in an expression such as x/(1-x) - x^2/(1-x): We simply simplify the expression to (x-x^2)/(1-x)=x before taking the limit, and get the perfectly well-defined value 1.

For simplicity, let me restrict to the case of a single massive Hermitian scalar field Phi(x) only - everything extends without difficulties to arbitrarily massive fields of arbitrary spin, and most of it applies also to massless fields (which may have additional infrared complications, though).

Writing Phi(f)=integral dx f(x) Phi(x) for arbitrary test functions f(x) (from Schwartz space),
one can define the multilinear Wightman functionals
$$W(f_1,...,f_N)=\langle\Phi(f_1)\cdots\phi(f_N)\rangle,$$
where the expectation is with respect to the vacuum state of the theory. W(f_1,...,f_N) can be written as a formal integral over Wightman distributions W(x_1,...,x_n), which are the limiting cases when the f_k tend to delta distributions centered at x_k. The formal properties of the Wightman functions of massive fields are expressed by the so-called Wightman axioms, http://en.wikipedia.org/wiki/

The Wightman axioms can be derived at the level of rigor conventional in theoretical physics for all renormalizable quantum field theories with a Poincare-invariant action. (In dimensions d<4, there are also mathematically fully rigorous constructions of interacting quantum field theories derived from a large class of Poincare-invariant actions; this branch of mathematical physics is called constructive field theory. But in the most important dimension d=4, there are technical obstacles that haven't been overcome so far in full rigor. Therefore, in this post, I shall argue only on the level of rigor as defined, e.g., by Weinberg's QFT treatise.) For an important step in this derivation, see the thread https://www.physicsforums.com/showthread.php?t=388556 .

From the Wightman functions, one can construct their time-ordered version and from these the usual S-matrix elements. However, one can do much more!

Given Wightman distributions satisfying the Wightman axioms, it is not difficult to construct the physical Hilbert space. It consists of all limits of linear combinations of terms |f_1,...,f_N> with Inner product
$$|\langle<g_1,...,g_M|f_1,...,f_N\rangle:=\langle\Phi(g_M)\cdots\Phi(g_1)\Phi(f_1)\cdots\phi(f_N)\rangle.$$
It is an instructive exercise to show that this indeed defines a Euclidean space whose closure is the required Hilbert space.

This Hilbert space carries a unitary representation of the Poincare group, in which the group element U acts as
$$|U|f_1,...,f_N\rangle:=|Uf_1,...,Uf_N\rangle$$
where Uf is the action of the Poincare group on the single particle space. In particular, the time translations form a 1-parameter group whose infinitesimal generator H is (by standard functional analysis) a densely defined, self-adjoint linear operator. The Wightman axioms guarantee that the spectrum of H is nonnegative, and that |> (the case N=0 of |f_1,...,f_N>) is the unique pure state annihilated by H. This is the physical vacuum state. The physical 1-particle states are the states |f_1>.

Thus everything required by standard quantum mechanics is in place - except that the derivation in dimension d=4 is not fully rigorous, and that the massless case needs extra considerations (which figure under the heading ''infraparticles'').

The CTP formalism is simply a way to construct the Wightman functions and their time-ordered version in a nicely arranged way that makes it comparatively simple to derive quantum kinetic equations (which are the basis for the derivation of semiconductor equations, hydrodynamic equations, etc.). See http://arxiv.org/pdf/hep-th/9504073 for an introduction, and Phys. Rev. D 37, 2878–2900 (1988) for a derivation of the Boltzmann equation.

 

[meopemuk]

It doesn't do without a Hamiltonian - only without a Hamiltonian expressed explicitly in terms of creation and annililation operators. Explaining this takes some preparations:

Your presentation was rather abstract, and I would like to see some concrete examples. Can you write the "CTP Hamiltonian" for a simple theory, like $$\phi^4$$, explicitly. Will it be the same Hamiltonian as (4.12) in Peskin-Schroeder? or more like the Hamiltonian with counterterms (10.18) in the same book? Or the "CPT Hamiltonian" is something completely different?

Can I use this Hamiltonian to solve some simple dynamical problems, like taking two particles at time 0 and see how their wave function evolves to a different time? I couldn't find answers to such simple questions in your references on kinetic equations.

Thanks.
Eugene.

 

[A. Neumaier]

Your presentation was rather abstract, and I would like to see some concrete examples. Can you write the "CTP Hamiltonian" for a simple theory, like $$\phi^4$$, explicitly.

One can write down the Hamiltonian explicitly, for arbitrary field theories defined by a covariant local action; see below. What one cannot write down explicitly (except to low order in perturbation theory) is the inner product.

Will it be the same Hamiltonian as (4.12) in Peskin-Schroeder? or more like the Hamiltonian with counterterms (10.18) in the same book? Or the "CPT Hamiltonian" is something completely different?

Something completely different, nonperturbatively defined on a non-Fock space. And it does not deserve to be called the CTP-Hamiltonian. The Hamiltonian itself is completely independent of the technique used to evaluate the Wightman function. If you need a name, call it the Wightman Hamiltonian.

Can I use this Hamiltonian to solve some simple dynamical problems, like taking two particles at time 0 and see how their wave function evolves to a different time? I couldn't find answers to such simple questions in your references on kinetic equations.

I specified the Hamiltonian completely. In particular, in a covariant position representation of the single particle space, the dynamics is very simple and explicit:The 2-particle wave function |psi(x_1,x_2)> at time t=0 evolves to |psi(x_1+tu,x_2+tu)> at time t (where u=(1,0,0,0^T). The only nontrivial thing is the inner product, which for an interacting theory is very different from a Fock inner product, since it requires the knowledge of the Wightman functions - which are computed approximately via CTP. Only in the free case, the Wightman functions can be written down explicitly, and correspond to a Fock inner product.

While you create a complicated dynamics on an inadequate Fock space, defined by an approximately known Hamiltonian, Wightman creates a simple dynamics in a complicated Hilbert space, defined by an approximately known metric (except in 2D, where there are a fair number of explicitly solvable models).

 

[meopemuk]

I specified the Hamiltonian completely. In particular, in a covariant position representation of the single particle space, the dynamics is very simple and explicit:The 2-particle wave function |psi(x_1,x_2)> at time t=0 evolves to |psi(x_1+tu,x_2+tu)> at time t (where u=(1,0,0,0^T). The only nontrivial thing is the inner product, which for an interacting theory is very different from a Fock inner product, since it requires the knowledge of the Wightman functions - which are computed approximately via CTP. Only in the free case, the Wightman functions can be written down explicitly, and correspond to a Fock inner product.

While you create a complicated dynamics on an inadequate Fock space, defined by an approximately known Hamiltonian, Wightman creates a simple dynamics in a complicated Hilbert space, defined by an approximately known metric (except in 2D, where there are a fair number of explicitly solvable models).

I would like to see how this "Wightman Hamiltonian" theory reduces to well-known and testable approximations for few-particle systems, like the hydrogen atom or clasical interaction dynamics of two charges, or something of that sort. I understand that currently this cannot be done with full mathematical rigor. But did anybody tried to introduce there some reasonable approximations, like perturbation theory or (v/c) expansion, and demonstrate how this approach is similar/different from well-tested QM used in atomic or molecular physics? At some point these "complicated Hilbert spaces" and "approximately known metrics" should reduce to something more familiar. Has anybody studied this transition?

For example, in my understanding, the non-Fock structure of the Hilbert space means that pure interacting 2-particle systems do not exist, because there is always some admixture of states with higher numbers of particles. How these admixture coefficients depend on the strength of interaction? Is it possible to see these n-particle components in experiments? What is the impact of these n-particle components on the energy spectrum of, say, hydrogen atom?

In my opinion, one advantage of the dressed particle dynamics in the "inadequate" Fock space is that transition to the ordinary quantum mechanics and classical mechanics of charges is simple and transparent.

Eugene.

 

[meopemuk]

Will it be the same Hamiltonian as (4.12) in Peskin-Schroeder? or more like the Hamiltonian with counterterms (10.18) in the same book? Or the "CPT Hamiltonian" is something completely different?

Something completely different, nonperturbatively defined on a non-Fock space. And it does not deserve to be called the CTP-Hamiltonian. The Hamiltonian itself is completely independent of the technique used to evaluate the Wightman function. If you need a name, call it the Wightman Hamiltonian.

Isn't it strange that the same theory uses two different things for the same purpose? I mean, there is Peskin-Schroeder Hamiltonian and there is Wightman Hamiltonian. Both of them can be used (presumably) to calculate the time evolution of states, and both of them can be used (presumably) to calculate the S-matrix. So, the natural question is how these two Hamiltonians are related to each other? Is one of them an approximation for the other? Or they are the same thing, but written differently? Are the calculated results the same or different?

Did somebody analyze these questions in simple low-dimensional models, where, as I understand, everything can be done rigorously and exactly?

Eugene.

 

[A. Neumaier]

Isn't it strange that the same theory uses two different things for the same purpose?

No. This is the typical superiority of abstract mathematics over pedestrian approaches, in all applications of mathematics to science and engineering.The abstract setting gives a much more global perspective for seeing what is going on, though on the surface things there often look very different.

I mean, there is Peskin-Schroeder Hamiltonian and there is Wightman Hamiltonian. Both of them can be used (presumably) to calculate the time evolution of states, and both of them can be used (presumably) to calculate the S-matrix. So, the natural question is how these two Hamiltonians are related to each other?
Is one of them an approximation for the other?

It is strange only at first sight. All Hamiltonians describe the same system as long as they are unitarily equivalent. This is why there is lots of freedom in similarity renormalization.

Peskin/Schroeder's, Dyson's, and your scheme are just particular (approximate) realizations of the latter. The Hamiltonian view on a Fock space given by Peskin & Schroeder is valid only with a cutoff in place (so that there are not yet any infinities), and therefore only approximates the true, renormalized theory. The Hamiltonian views given by Dyson and you are only valid up to a certain order, and hence also approximate.
In all three cases, the existence of the limit is very problematic, though the problems show in different ways in different approaches.

But within the approximation, one can always go from the Hamiltonian view (which represents the states by means of field operators at fixed time) to the Wightman view by defining the interactive fields to be the covariant fields obtained by conjugating the fields at the space-time origin with exp(ix dot P), where P is the momentum operator of the approximate representation of the Poincare group, and taking vacuum expectation values of their products. The interesting thing is that in the Wightman formulation, Fock space disappeared, since the Hilbert space is constructed from the Wightman functions rather than given a priori. Therefore Haag's obstacle to forming the right limit disappeared in this formulation. Indeed, the limit exists in Wightman's setting in all cases where the limit proved tractable at all so far.

Since the limiting Hilbert space is no longer a Fock space, the reverse path - going from the Wightman formulation to a Hamiltonian formulation in Fock space, doesn't work. Constructive field theory provides sometimes more explicit Hamiltonian formulations, but these are not really intelligible enough (to me, at least) for doing actual calculations.

Or they are the same thing, but written differently? Are the calculated results the same or different?

If the limit is done at the end of the calculations (after all potential infinities have been cancelled), the results should be the same.

Did somebody analyze these questions in simple low-dimensional models, where, as I understand, everything can be done rigorously and exactly?

I am working on understanding this myself, and haven't yet gotten very far. I have been trying to find papers that give explicit N-point Wightman functions for some of the exactly solvable models, but haven't been successful so far. The language of those who analyze exactly solvable models is so different from what I am familiar with that I have only a superficial understanding of what was they do means in terms of the concepts I am familiar with. So it is really hard work for me to interpret what they do. Moreover, the language seems to have shifted completely in the 1980s, and the substance is now somewhere hidden in the very voluminous work on quantum groups. Maybe DarMM can help out...

 

[A. Neumaier]

I would like to see how this "Wightman Hamiltonian" theory reduces to well-known and testable approximations for few-particle systems, like the hydrogen atom or clasical interaction dynamics of two charges, or something of that sort. I understand that currently this cannot be done with full mathematical rigor. But did anybody tried to introduce there some reasonable approximations, like perturbation theory or (v/c) expansion, and demonstrate how this approach is similar/different from well-tested QM used in atomic or molecular physics? At some point these "complicated Hilbert spaces" and "approximately known metrics" should reduce to something more familiar. Has anybody studied this transition?

I'd like to see this, too. But those working on axiomatic/constructive field theory are far too abstract for my taste, and seem not to be much interested in this sort of questions.
So these seems to be an open research topic about which one could publish partial results.

For example, in my understanding, the non-Fock structure of the Hilbert space means that pure interacting 2-particle systems do not exist,

But they don't even exist in your picture, since at high energies, the S-matrix for two entering particles produces arbitrarily many leaving particles. The hydrogen atom as a 2-particle system is already an idealization!

 

[meopemuk]

The language of those who analyze exactly solvable models is so different from what I am familiar with that I have only a superficial understanding of what was they do means in terms of the concepts I am familiar with. So it is really hard work for me to interpret what they do.

How true! I gave up the idea to read and understand papers in axiomatic QFT. I just wish these people spend some time to consider simple 2-particle models and explain us mortals how the non-Fock space is organized.

Eugene.

 

[meopemuk]

I'd like to see this, too. But those working on axiomatic/constructive field theory are far too abstract for my taste, and seem not to be much interested in this sort of questions.

Exactly!

But they don't even exist in your picture, since at high energies, the S-matrix for two entering particles produces arbitrarily many leaving particles. The hydrogen atom as a 2-particle system is already an idealization!

It is true that in the Fock space picture one can form states, which are linear superpositions of n-particle states with various n. For example, the hydrogen atom can be seen as primarily electron+proton plus admixture of higher-n states. However, it is important that one can always form states in which the number of particles is determined exactly. These states are likely to be non-stationary, but they are present, nevertheless.

On the other hand, it seems that non-Fock spaces do not allow such a luxury as a state with a well-defined number of particles.

Eugene.

 

[A. Neumaier]

It is true that in the Fock space picture one can form states, which are linear superpositions of n-particle states with various n. For example, the hydrogen atom can be seen as primarily electron+proton plus admixture of higher-n states. However, it is important that one can always form states in which the number of particles is determined exactly. These states are likely to be non-stationary, but they are present, nevertheless.

Yes, but they don't qualify for what you had asked for: ''pure interacting 2-particle systems''. Once you have nontrivial scattering, a pure 2-particle system can exist only for a fleeting moment (and in practice never, except at t=-inf).

On the other hand, it seems that non-Fock spaces do not allow such a luxury as a state with a well-defined number of particles.

In a massive Wightman field theory, there is a well-defined vacuum state Omega, and all states a^*(f)Omega qualify for being the interacting analogue of single particle states.
Product states such as a^*(f)a^*(g)Omega can still be viewed as the interacting analogue of 2-particle states (which they are in the free case). However, in a fixed reference frame (of the center of mass of the scattering system, where a Hamiltonian picture makes sense), these 2-particle states are no longer composed of exactly two in/out particles, giving rise to nontrivial scattering.

 

[A. Neumaier]

How true! I gave up the idea to read and understand papers in axiomatic QFT. I just wish these people spend some time to consider simple 2-particle models and explain us mortals how the non-Fock space is organized

They cannot, for the reasons given in the other mail.

But I agree that there is insufficinet interfacing information between ordinary QFT and the axiomatic version. The lack of this makes reading for newcomers much more difficult than necessary.

 

[DarMM]

Regarding the difference between the Wightman Hamiltonian and the normal (Peskin and Schroeder) Hamiltonian, it is a bit subtle.

The Peskin and Schroeder Hamiltonian is a formal operator on Fock space, it isn't really an operator on Fock space, but you right it down as if it is. The Wightman Hamiltonian is a well-defined operator on a non-Fock space. They have the same form, i.e., they are the same function of the field, the difference is the space you right them on.

Of course in standard treatments you continue to use the Hamiltonian on Fock space and run into ultraviolet divergences, these can be cured by adding very singular operators to the Hamiltonian. After doing this the Hamiltonian is still not a well-defined operator. However it is good enough for the perturbative expansion of the S-matrix, the formula for the perturbative S-matrix is now well-defined. However finite time evolution and the non-perturbative S-matrix are still an issue.

So we original have the Hamiltonian, an abstract function of the field. To implement it, we need to pick a Hilbert space, or in other terms pick an expression for the field. In Wightman approach you pick the right Hilbert space at the start or another way to put it you pick the correct functional form for the field. The theory is then completely finite.
In the standard approach you work on Fock space, however Fock space only works for free theories so you encounter divergences. You cannot obtain a well-defined Hamiltonian (impossible on Fock space by Haag's theorem), however you can add singular operators to the Hamiltonian and obtain a perturbatively well-defined S-matrix and n-point functions.



The connection between the Wightman formalism and the standard formalism has been given in a few papers.
In two dimensions:
For example the fact that the taylor expansion of the Wightman functions in the non-Fock space agree with renormalized perturbation theory can be found in:
Dimock J., "Asymptotic perturbation expansion in the $$\mathcal{P}(\phi)_{2}$$ quantum field theory", Comm. Math. Phys., 35, 347 - 356

Also the perturbative expansion of the S-matrix on the non-Fock space agrees with renormalized perturbation theory. This is proven in:
Osterwalder K. and Sénéor R., "A nontrivial scattering matrix for weakly coupled $$\mathcal{P}(\phi)_{2}$$ models", Helv. Phys. Acta., 49, 525-535.

Also Borel summation of the renormalized perturbation theory gives the Wightman functions on the non-Fock space. This is proven in:
Sokal A. "An improvement of Watson's theorem on Borel summability", J. Math. Phys. 21, 261-263

In three dimensions, taylor expansion of the non-Fock Wightman functions agrees with renormalized perturbation theory and Borel summation of renormalized perturbation theory gives the non-Fock Wightman functions. Proof found in:
Magnen J. and Sénéor R. "Phase cell expansion and Borel summability for the Euclidean theory", Comm. Math. Phys, 56, 237-276

This is also true of the S-matrix:
Constantinescu F., "Nontriviality of the scattering matrix for weakly coupled $$\phi^{4}_{3}$$ models" Ann. Physics, 108, 37-48


The simplest example of explicit n-point functions is usually found in work on the Thirring model. For example:
D. Ts. Stoyanov and L. K. Khadzhiivanov, "Theory of Wightman functions in the thirring model", Theoretical and Mathematical Physics 46, 3, 236-242
This is because the Thirring model has a rigorously defined transformation to a free scalar field, so you obtain the free scalar correlation functions and then transform back to the Thirring model. If anybody would like more difficult 2d models please say so.

 

[A. Neumaier]

Regarding the difference between the Wightman Hamiltonian and the normal (Peskin and Schroeder) Hamiltonian, it is a bit subtle.

The Peskin and Schroeder Hamiltonian is a formal operator on Fock space, it isn't really an operator on Fock space, but you right it down as if it is. The Wightman Hamiltonian is a well-defined operator on a non-Fock space. They have the same form, i.e., they are the same function of the field, the difference is the space you right them on.

But they can't have the same coefficients, since these are divergent in the P/S Hamiltonian. Do you mean that they can be written as the same limit, if one uses the right cutoff-dependent counterterms?

So we original have the Hamiltonian, an abstract function of the field. To implement it, we need to pick a Hilbert space, or in other terms pick an expression for the field. In Wightman approach you pick the right Hilbert space at the start or another way to put it you pick the correct functional form for the field.

The start apparently is to assume having the right inner product? But this is not really known...

the perturbative expansion of the S-matrix on the non-Fock space agrees with renormalized perturbation theory. This is proven in:
Osterwalder K. and Sénéor R., "A nontrivial scattering matrix for weakly coupled $$\mathcal{P}(\phi)_{2}$$ models", Helv. Phys. Acta., 49, 525-535.

Thanks for the references. See also: http://arxiv.org/pdf/math-ph/0306042 for the Epstein-Glaser version of Phi^4_2.

The simplest example of explicit n-point functions is usually found in work on the Thirring model. For example:
D. Ts. Stoyanov and L. K. Khadzhiivanov, "Theory of Wightman functions in the thirring model", Theoretical and Mathematical Physics 46, 3, 236-242
This is because the Thirring model has a rigorously defined transformation to a free scalar field, so you obtain the free scalar correlation functions and then transform back to the Thirring model. If anybody would like more difficult 2d models please say so.

Thanks a lot. The other models later, after I understood this one...

For now, I have a different question: According to the general theory, the Phi(f) should be self-adjoint for real f, so U(f):=e^{i Phi(f)} should be well-defined. It seems to me that due to renormalization, <U(f)>=1 for all f -- is this correct? I wonder whether for some exactly solvable field theory there are closed formulas for <U(f_1)U(f_2)> and perhaps expectations of longer such products?

And another question: Where would one find all this information if one would have to search for oneself? I found the literature on these topics extremely difficult to access.
How did you learn it?

 

[DarMM]

But they can't have the same coefficients, since these are divergent in the P/S Hamiltonian. Do you mean that they can be written as the same limit, if one uses the right cutoff-dependent counterterms?

Only that they have the same rough functional form. In distribution theory infinite coefficients are standard (related to the problem of multiplication and the Hahn-Banach theorem, I will explain this eventually in the ladder tread and also when I get around to your "What is renormalization" thread). Hence from a distributional point of view they are the same abstract algebraic object, waiting to be made concrete by a choice of representation for the field.

The start apparently is to assume having the right inner product? But this is not really known...

Yes, or the equivalently the correct distributional expression for the field. These are not known, so in a constructive approach you really find the Hilbert space through renormalization.

According to the general theory, the Phi(f) should be self-adjoint for real f, so U(f):=e^{i Phi(f)} should be well-defined. It seems to me that due to renormalization, <U(f)>=1 for all f -- is this correct? I wonder whether for some exactly solvable field theory there are closed formulas for <U(f_1)U(f_2)> and perhaps expectations of longer such products?

When you say, <U(f)>=1 for all f, is the expectation taken in the vacuum?

The expectations and existence of objects like U(f) and their products is basically related to verifying the Haag-Kastler axioms. The abstract C*-algebra is composed from the U(f). It was easier to prove the existence, causality, e.t.c. of these objects first, which is why the Haag-Kastler axioms were actually proven for $$\mathcal{P}(\phi)_{2}$$ before the Wightman axioms. (By before I mean a few months before).

And another question: Where would one find all this information if one would have to search for oneself? I found the literature on these topics extremely difficult to access.
How did you learn it?

It is extremely difficult to access. I essentially read Glimm and Jaffe and PCT, Spin and statistics and kept reading references based on questions they generated. I basically wrote notepad after notepad of notes, cross-referencing papers and every time a question occurred to me I searched everywhere I knew to find the answer. This often involved making notes on what authors said their references demonstrated so that I could quickly find the right paper. Basically I climbed the ladder with a sequence of questions for every rung, crossing it off when I had read the appropriate paper. There is no easy way to do it because there is no good monograph on the subject for an outsider (I hope to write one). For instance Glimm and Jaffe's textbook is full of comments that will not make sense unless you have read other papers and it will not leave you with a sense of what questions are important.

 

[A. Neumaier]

When you say, <U(f)>=1 for all f, is the expectation taken in the vacuum?

Yes, as for the Wightman functions themselves. But I am guessing from half-understood material...

The expectations and existence of objects like U(f) and their products is basically related to verifying the Haag-Kastler axioms. The abstract C*-algebra is composed from the U(f). It was easier to prove the existence, causality, etc. of these objects first

This gives existence, but to get a better feeling for what is going on, I'd like to be able to play with explicit formulas in some nontrivial example(s).

It is extremely difficult to access. I essentially read Glimm and Jaffe and PCT, Spin and statistics and kept reading references based on questions they generated. I basically wrote notepad after notepad of notes, cross-referencing papers and every time a question occurred to me I searched everywhere I knew to find the answer.

I proceed like that, with the same books to start with (together with Reed & Simon) but at a much slower pace, since I have so many other duties and interests. So my views on this are much more sketchy than yours and at times vague.

This often involved making notes on what authors said their references demonstrated so that I could quickly find the right paper. Basically I climbed the ladder with a sequence of questions for every rung, crossing it off when I had read the appropriate paper. There is no easy way to do it because there is no good monograph on the subject for an outsider (I hope to write one).

To make sure that your future monograph is at least readable by a half-outsider, I offer to proofread parts of your future draft, at the level of my understanding!

 

[A. Neumaier]

I just wish these people spend some time to consider simple 2-particle models and explain us mortals how the non-Fock space is organized.

Fock space already comprises N-particle dynamics for all N; and non-Fock space is not different.

http://www.springerlink.com/content/3nmk556771471575/ (chapter 3 of the book https://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=3540735925&tag=pfamazon01-20 ) gives a concrete idea of how non-Fock representations arise.

 

[A. Neumaier]

For example, the hydrogen atom can be seen as primarily electron+proton plus admixture of higher-n states. However, it is important that one can always form states in which the number of particles is determined exactly.

Why should that be important? These states are nonobservable, anyway!

 

[A. Neumaier]

The simplest example of explicit n-point functions is usually found in work on the Thirring model. For example:
D. Ts. Stoyanov and L. K. Khadzhiivanov, "Theory of Wightman functions in the thirring model", Theoretical and Mathematical Physics 46, 3, 236-242
This is because the Thirring model has a rigorously defined transformation to a free scalar field, so you obtain the free scalar correlation functions and then transform back to the Thirring model. If anybody would like more difficult 2d models please say so.

I'd like to see the explicit the n-point functions (and their unitary C^* algebra counterpart, and renormalized field equations and commutation relations - as far as available) for 2-dimensional QED.

 

[meopemuk]

Why should that be important? These states are nonobservable, anyway!

I think that they are observable, at least, in principle. It is true that once prepared such states immediately transform into superpositions of n-particle states, so it is difficult to observe an interacting state with a well-defined number of particles. Nevertheless, such states exist in the Fock-space theory, and given enough instrumental resolution we should be able to distinguish such states. So, in principle, experiments can tells us whether Fock-space or non-Fock-space is realized in nature. I wonder if somebody has analyzed the experimental differences between the two approaches?

Eugene.

 

[A. Neumaier]

I think that they are observable, at least, in principle.

Then please propose a gedanken experiment to observe them.

Nevertheless, such states exist in the Fock-space theory, and given enough instrumental resolution we should be able to distinguish such states.

You argue that ''it is important that one can always form states in which the number of particles is determined exactly'' in order to justify your preference for Fock space, and ''such states exist in the Fock-space theory, [so] we should be able to distinguish such states''. This is called circular reasoning.

 

[DarMM]

So, in principle, experiments can tells us whether Fock-space or non-Fock-space is realized in nature. I wonder if somebody has analyzed the experimental differences between the two approaches?

A non-Fock space is not another alternate theory to QFT, it's standard QFT made rigorous. By Haag's theorem QFTs which interact have to live in a non-Fock space, a perturbative version of their S-matrix can exist in Fock space (which is why non-rigorous methods work), but nothing more.

A theory which used Fock spaces for relativistic interactions would not be a field theory. So your questions concerns the difference between field theory and some other formalism. They would not be different approaches to field theory, but totally different formalisms.

 

[meopemuk]

You argue that ''it is important that one can always form states in which the number of particles is determined exactly'' in order to justify your preference for Fock space, and ''such states exist in the Fock-space theory, [so] we should be able to distinguish such states''. This is called circular reasoning.

Let me be clear. I meant that in a Fock theory one can always form states in which the number of particles is determined exactly. In a non-Fock theory such states do not exist. At this point I am not saying which of the two approaches is better. I am simply stating that they are different and that this difference can be seen in very sensitive experiments.

The relevant experiment should be able to tell exactly how many particles there are in a given state. In practice this is not possible. For example, soft photons always escape detection.

Eugene.

 

[A. Neumaier]

Let me be clear.[...] this difference can be seen in very sensitive experiments.

The relevant experiment should be able to tell exactly how many particles there are in a given state. In practice this is not possible.

That what is not possible in practice can be seen in very sensitive experiments sounds contradictory rather than clear.

In any case, this has nothing to do anymore with the thread.

 

[meopemuk]

A theory which used Fock spaces for relativistic interactions would not be a field theory. So your questions concerns the difference between field theory and some other formalism. They would not be different approaches to field theory, but totally different formalisms.

I agree completely. There are two approaches to relativistic quantum theory.

One is the field-based approach. In this approach people postulate that there exist covariant interacting quantum fields with zero space-like (anti)commutators. Under this assumption one proves Haag's theorem and is forced to work in non-Fock spaces.

The other approach does not postulate the existence of covariant interacting quantum fields with zero space-like commutators. This approach does not pay attention to fields at all. It is based on the particle interpretation. Examples are provided by "dressed particle" theories of Greenberg-Schweber. These theories satisfy all physical principles, such as Poincare commutators and cluster separability. Since conditions of Haag's theorem do not hold, the theorem itself is not applicable, and one is free to work in the Fock space. See

M. I. Shirokov, "Dressing" and Haag's theorem, http://arxiv.org/abs/math-ph/0703021

So, as I said elsewhere, we have two completely different theories, which are (possibly) internally self-consistent. The only way to say which one is right and which one is wrong is to compare their predictions with experiments.

There is a similar discussion in the parallel thread https://www.physicsforums.com/showthread.php?p=3164679

Eugene.

 

[A. Neumaier]

One is the field-based approach. [...]
The other approach [...] does not pay attention to fields at all.

So the other approach and its assumptions should not be discussed in this thread, which is about ''Time evolution in quantum field theories''.

Instead of trying to learn more about standard quantum field theories (and how they handle particles as approximate concepts), you try to discuss your personal particle theories (in which fields only have a slave's position) for which you have your own thread.

 

[A. Neumaier]

I would like to see how this "Wightman Hamiltonian" theory reduces to well-known and testable approximations for few-particle systems, like the hydrogen atom or classical interaction dynamics of two charges, or something of that sort.

OK, let us take an anharmonic oscillator - this is simpler than the hydrogen atom, which has an electron in a Coulomb potential: Scattering states are absent, and neither spin nor angular momentum need to be accounted for.

I'll guide you into doing the computations yourself.

Consider the anharmonic oscillator with Hamiltonian H=p^2/2m+V(q), where
V(q)=kq^2/2+aq^3/3+bq^4/4
and b>0, acting in the standard Fock representation with a single mode. The anharmonic oscillator is a 1+0 dimensional QFT in which space consists of a single point only (namely the oscillating mode), and everything happens in the simplest Fock space F, that you understand very well.

Let q(t) be the time-dependent position operator in the Heisenberg picture, and define the Wightman functions
W(t_1,...,t_N):=Omega^*q(t_1)...q(t_N)Omega,
where Omega is the ground state of the Hamiltonian, N=0,1,2,..., and the t_k are arbitrary times.

Please work out q(t), Omega, and W(t_1,...,t_N) in perturbation theory to the least nontrivial order, assuming that a, b are tiny. If you like, you may put m=1, k=1, and either a=0 or b=0 to simplify the formulas. (b=0 is simpler than a=0 but is a nonphysical limiting case.)

If this is sucessfully completed, I'll tell you how to use the Wightman functions.

 

[meopemuk]

Consider the anharmonic oscillator with Hamiltonian H=p^2/2m+V(q), where
V(q)=kq^2/2+aq^3/3+bq^4/4
and b>0, acting in the standard Fock representation with a single mode. The anharmonic oscillator is a 1+0 dimensional QFT in which space consists of a single point only (namely the oscillating mode), and everything happens in the simplest Fock space F, that you understand very well.

Let q(t) be the time-dependent position operator in the Heisenberg picture, and define the Wightman functions
W(t_1,...,t_N):=Omega^*q(t_1)...q(t_N)Omega,
where Omega is the ground state of the Hamiltonian, N=0,1,2,..., and the t_k are arbitrary times.

Please work out q(t), Omega, and W(t_1,...,t_N) in perturbation theory to the least nontrivial order, assuming that a, b are tiny. If you like, you may put m=1, k=1, and either a=0 or b=0 to simplify the formulas. (b=0 is simpler than a=0 but is a nonphysical limiting case.)

First I would like to ask you about the terminology. Why do you call this a 1+0 dimensional QFT? This looks more like an example in ordinary 1-dimensional 1-particle quantum mechanics. Also I don't see how "space consists of a single point only ". I understand that in this example q is the space coordinate which varies from -infinity to +infinity.

Anyway, since you gave an explicit Hamiltonian, this case should be fully solvable. To find the new ground state I would solve the stationary Schroedinger equation

H |Omega> = E |Omega>

In principle, this can be solvable by perturbation theory. However, one should be careful as even for small a and b the minimum of the new potential can be very far from q=0, so that the perturbation expansion may not be well-converging. The time evolution of the position operator is given by the usual formula

q(t) = exp(iHt) q(0) exp(-iHt)

With enough computational power this operator can be calculated with sufficient precision for any t, and I don't see a problem of calculating the Wightman function

W(t_1,...,t_N):= <Omega| q(t_1)...q(t_N)|Omega>

What's next?
Eugene.

 

[A. Neumaier]

First I would like to ask you about the terminology. Why do you call this a 1+0 dimensional QFT? This looks more like an example in ordinary 1-dimensional 1-particle quantum mechanics.

To convince yourself of the correctness of my interpretation, reduce the description of Weinberg (5.1.1) to the case of a point (zero space dimension), and you'll find that the free field in 1+0 dimensions is just the harmonic oscillator. The Poincare group has dimension 1 and is generated by P_0=H. This is the reason why everything is very simple, no representation theory is needed! This is why I start with this extreme case. This is a warm-up exercise; we'll do 1+3 dimensions after enough ground has been prepared.

Also I don't see how "space consists of a single point only ". I understand that in this example q is the space coordinate which varies from -infinity to +infinity.

You need to rethink all you know about the anharmonic oscillator in the new light of my explanations.

Anyway, since you gave an explicit Hamiltonian, this case should be fully solvable. To find the new ground state I would solve the stationary Schroedinger equation
H |Omega> = E |Omega>

I thought that you'd actually do it to lowest nontrivial order, in order to appreciate what happens. Remember: You must pay for my efforts with yours!

In principle, this can be solvable by perturbation theory. However, one should be careful as even for small a and b the minimum of the new potential can be very far from q=0, so that the perturbation expansion may not be well-converging.

In this case, you really _need_ to do the exercise explicitly in some detail. It will save you from making such misassessments. Take a,b as fixed multiples of a sufficiently small coupling constant g.

The time evolution of the position operator is given by the usual formula
q(t) = exp(iHt) q(0) exp(-iHt)

Yes. Please evaluate that to lowest nontrivial order.

With enough computational power this operator can be calculated with sufficient precision for any t,

For the purpose of the exercise, it is enough doing it at lowest nontrivial order. Then the computational power of a student doing QM exercises by hand is enough.

and I don't see a problem of calculating the Wightman function
W(t_1,...,t_N):= <Omega| q(t_1)...q(t_N)|Omega>

Then do it. For our purposes, it is enough to do the cases N<=4 to lowest nontrivial order.

What's next?

The next thing is to define the vectors
|t_1,...,t_N> := q(t_1)...q(t_N)|Omega>
and evaluate them to lowest nontrivial order. To look further ahead, reread my older messages in this thread; but I'll repeat everything as we go along.

 

[meopemuk]

To convince yourself of the correctness of my interpretation, reduce the description of Weinberg (5.1.1) to the case of a point (zero space dimension), and you'll find that the free field in 1+0 dimensions is just the harmonic oscillator. The Poincare group has dimension 1 and is generated by P_0=H. This is the reason why everything is very simple, no representation theory is needed! This is why I start with this extreme case. This is a warm-up exercise; we'll do 1+3 dimensions after enough ground has been prepared.

OK, so you've started from a very unrealistic example, where the space consists of just one point, and in this point particles can be created and annihilated. So, there are c/a operators a* and a defined at this point. Your p and q have absolutely no relationship to physical momenta and positions. They are just linear combinations of a* and a that happen to satisfy the same commutators as physical momenta and positions.

The next thing is to define the vectors
|t_1,...,t_N> := q(t_1)...q(t_N)|Omega>
and evaluate them to lowest nontrivial order. To look further ahead, reread my older messages in this thread; but I'll repeat everything as we go along.

Can we shorten this exercise? I just want to see where we are going with this? I agree that vectors |t_1,...,t_N> := q(t_1)...q(t_N)|Omega> can be computed with sufficient accuracy. Let us skip these boring calculations and assume that we got the vectors. How does this bring us closer to the time evolution of multiparticle states in the Fock space?

Eugene.

 

[A. Neumaier]

OK, so you've started from a very unrealistic example, where the space consists of just one point, and in this point particles can be created and annihilated. So, there are c/a operators a* and a defined at this point. Your p and q have absolutely no relationship to physical momenta and positions. They are just linear combinations of a* and a that happen to satisfy the same commutators as physical momenta and positions.

One shouldn't make life more difficult than necessary. People study in class the particle in a box, the anharmonic oscillator, the hydrogen atom, the Dirac equation in an external field, and other simple things precisely because of that - to get practice without all the complexity that a real life example has.

Using this example, one can learn almost everything about the Wightman representation one needs to understand . And my time is precious; so I don't want to waste it on more complexity than is essential.

Can we shorten this exercise? I just want to see where we are going with this? I agree that vectors |t_1,...,t_N> := q(t_1)...q(t_N)|Omega> can be computed with sufficient accuracy. Let us skip these boring calculations and assume that we got the vectors.

Well, if you are impatient, simply apply the recipes I gave earlier to any example of your choice. I told you already everything needed.

But if you want to have my guidance and correction, follow the path I find necessary to set up the demonstration material and to get the practice needed. You don't need to write down the intermediate steps, but we need the formulas that result.

You can take m=k=omega to get rid of all square roots, and if you are lazy you can set omega=1 and either a=0 or b=0 to simplify things. (If b=0, which simplifies more, the potential is unbounded below but this doesn't matter in perturbation theory.)

How does this bring us closer to the time evolution of multiparticle states in the Fock space?

Ah; I forgot to explain:This _is_ already a multiparticle situation in our baby Fock space. The single particle Hilbert space is just the space of complex numbers. The ground state of the anharmonic oscillator is the vacuum state, and the k-th excited state is the k-particle state in 1+0 dimensions. You can easily verify this by reducing Weinberg's 1+3 dimensional treatmend to the case of a single space point.

 

[meopemuk]

Arnold,

before doing any specific exercises I would like to understand the logic of what's going on.

It seems that we both agree that we are doing quantum mechanics in a certain Hilbert space (Fock space or non-Fock space, whatever). One of quantum postulates is that the time evolution is described by the Hamiltonian H as

$$|\psi(t) \rangle = \exp(iHt) |\psi(0) \rangle$$......(1)

If I understand correctly, the axiomatic Wightman approach does not use this formula and even refuses to provide an explicit form of the Hamiltonian H. Instead, it is suggested to use a set of Wightman functions in order to do the time evolution. I can believe that such Wightman functions can be calculated (perturbatively or non-perturbatively, exactly or approximately, it doesn't matter now). I can even believe that using these Wightman functions one can reproduce the same time evolution as given by equation (1).

If this is so, then given the full set of Wightman functions one should be able to recreate the Hamiltonian H. Presumably, this Hamiltonian should be formulated in terms of observables of physical (rather than bare) particles. I would like to know whether such a Hamiltonian has been constructed in simple QFT models? I haven't seen explicit formulas in the references that you've provided earlier. Is it because

(a) the Hamiltonian formulation is theoretically unacceptable for some reason?

(b) this is difficult to do, but people are working on it?

(c) I've just missed this piece of information?

(d) people working on these models dislike Hamiltonians for some reason, or just don't care to derive them?

(e) other?

Thanks.
Eugene.

 

[A. Neumaier]

It seems that we both agree that we are doing quantum mechanics in a certain Hilbert space (Fock space or non-Fock space, whatever). One of quantum postulates is that the time evolution is described by the Hamiltonian H as

$$|\psi(t) \rangle = \exp(iHt) |\psi(0) \rangle$$......(1)

Yes. This is the meaning of the statement that H=P_0 is the generator of time translations. It is as true in QFT as in QM, and therefore also holds in the Wightman representation.

If I understand correctly, the axiomatic Wightman approach does not use this formula and even refuses to provide an explicit form of the Hamiltonian H.

This is incorrect. I gave you the explicit formula

$$H \psi(x_1,t_1,...x_N,t_N) = -i\sum_k d/dt_k \psi(x_1,t_1,...x_N,t_N)$$

If this is so, then given the full set of Wightman functions one should be able to recreate the Hamiltonian H. [...] I would like to know whether such a Hamiltonian has been constructed in simple QFT models? I haven't seen explicit formulas in the references that you've provided earlier.

(c) I've just missed this piece of information?

(c) is correct. I just repeated the formula for your convenience.
To understand its meaning, I invite you to do the exercises I recommended.

 

[meopemuk]

This is incorrect. I gave you the explicit formula

(c) is correct. I just repeated the formula for your convenience.
To understand its meaning, I invite you to do the exercises I recommended.

It looks like we are going in circles, aren't we? Let me make another attempt. Suppose we are doing a low-dimensional theory (phi^4 or something else, doesn't matter). Suppose also that I've prepared a well-separated couple of particles at time 0. I have a wave function for this state. I would like to know how this wave function evolves in time. I know how to do that with the Hamiltonian expressed in terms of a/c operators of particles.

However, your Hamiltonian does not have this form. So, which steps should I take in order to obtain the wave function at time t by using the Hamiltonian you've wrote? I am not asking for exact and complete algorithm. I just want to understand the idea.

Eugene.

 

[A. Neumaier]

It looks like we are going in circles, aren't we?

We are going in circles because you misunderstand something that I can clarify only through worked examples.

However, your Hamiltonian does not have this form. So, which steps should I take in order to obtain the wave function at time t by using the Hamiltonian you've wrote?

You should do the exercises so that I can explain. I can't explain the later steps without the former steps done, because I can see already from the way you respond that you wouldn't understand.

 

[meopemuk]

You should do the exercises so that I can explain. I can't explain the later steps without the former steps done, because I can see already from the way you respond that you wouldn't understand.

In a different thread https://www.physicsforums.com/showthread.php?t=474666&page=8 we've agreed that our views on quantum mechanics are completely different. We even disagreed on whether the square of a wave function can be interpreted as a probability density. So, if you want to teach me something you'll need to do it starting from the kindergarten level, indeed. I don't think this is worth your time and effort.

Thank you.
Eugene.