What is "dressed particle" approach to QFT?

I promised to tell about it several times. Now is the time.

Recall our discussion of the QED Hamiltonian where we found that the presence of terms like a*c*a makes this Hamiltonian useless for studying the time evolution. The problem was that these terms act non-trivially on 1-particle states. The main idea of the "dressed particle" approach is that there can be no interaction in the vacuum state and in 1-particle states. These states should evolve in time as if there were no interaction at all. Mathematically this requirement means that all terms in the interaction Hamiltonian should have at least two annihilation operators. In order to maintain the Hermiticity, there should be at least two creation operators in each interaction term. So, in a theory of interacting electrons and photons, the simplest normally ordered terms allowed in the interaction Hamiltonian are

a*a*aa
a*c*ac
a*a*c*aa
a*a*a*aaa
...

The first term here describes a direct electron-electron interaction; the second term is an electron-photon potential responsible for the Compton scattering; the third term describes bremsstrahlung, i.e., emission of photons in electron-electron collisions; the fourth term is a three-body electron-electron-electron potential,...

It is important that all these terms yield zero when acting on the vacuum and 1-particle states. This means, for example, that the electron mass is not affected by interaction, so there is no need for renormalization.

As I mentioned here , a number of relativistically invariant and cluster separable models of this sort were constructed by H. Kita. There is also a "dressing approach" by Greenberg and Schweber which allows one to make a "dressed particle" theory out of virtually any usual QFT. What is really exciting is that this approach works and one can construct a "dressed" version of QED and other popular theories. This provides a completely new perspective on foundations of relativistic quantum physics.

Weinberg's book

If you are interested in QFT, I strongly recommend you to read Weinberg's book [1]. It is so much different from all other books on the subject. Weinberg does not pretend that the idea of quantum fields arises from "generalization" of quantum-mechanical wave functions or as a quantum version of classical field theory. In Weinberg's book you will not find rather dubious logical chains like "Schroedinger equation -> Dirac equation -> Dirac's hole theory -> QFT" or "Lagrangian -> Klein-Gordon equation -> quantization -> quantum fields". His approach to quantum fields, Lagrangians, and gauges is pretty utilitarian.

Weinberg correctly says that the only thing we are interested in QFT is the S-matrix. We want the S-matrix to be (at least) relativistically invariant and cluster separable. This implies that the Hamiltonian of the theory should be relativistically invariant and cluster separable. He goes on to demonstrate how Hamiltonians with these properties can be constructed as certain polynomials of operator functions defined on the Minkowski space-time. These operator functions are called quantum fields and they are required to satisfy some formal properties, like (anti-)commutativity at space-like intervals and covariant Lorentz transformation laws. The fact that quantum fields satisfy Klein-Gordon or Dirac equations is almost accidental, and has no any fundamental significance.

In Weinberg's logic, there is no reason to assume that quantum fields have any physical interpretation. They are just formal mathematical constructs. Moreover, there is no reason to associate the Minkowski space-time on which the fields "live" with real physical space and time.

This is by far the best book written about QFT. It gives you a very clear and, in my opinion, the only correct perspective on the foundations of QFT. Enjoy reading!

[1] S. Weinberg, The Quantum Theory of Fields, Vol. 1 , (University Press, Cambridge, 1995)

What are quantum fields?

Relativistic QFT is a fine theory. If you hold your nose and follow textbook recipes without asking too many questions, you can finally calculate a lot of stuff (scattering cross-sections, Lamb shifts, etc.) in a remarkable agreement with experiment.

However, on this blog we are going to ask questions. Why not? For example:

• Q: What are quantum fields?
• A: They are operator functions on the Minkowski space-time.
• Q: What is their physical meaning?
• A: They are obtained by second quantization of wave functions, at least in the non-interacting theory.
• Q: Wave functions have probabilistic interpretation, and in a relativistic theory the probabilities must be invariant with respect to Poincare transformations. Is it true?
• A: Yes.
• Q: Then particle wave functions must transform by a unitary representation of the Poincare group. Right?
• A: Right.
• Q: So, if quantum fields (even non-interacting) are somehow related to particle wave functions, they also must transform unitarily. Right?
• A: That sounds plausible.
• Q: However, this is not true. Take, for example, Dirac field for electrons. Lorentz transformations of this field are represented by 4x4 matrices. However, it is well-known that there are no finite-dimensional unitary representations of the non-compact Lorentz group. So, Lorentz transformations of the Dirac field are non-unitary.
• A: So what?
• Q: This means that there is no direct connection between the electron's quantum field and electron's wave function?
• A: I am not sure.
• Q: In the quantum field ψ(x,t) the argument x is physical position and t is time. Isn't it?
• A: Yes.
• Q: Is it possible to think (as we do in quantum mechanics) that x as an eigenvalue of the position operator?
• A: I guess so.
• Q: Then we should accept the existence of the time operator as well, which is impossible .
• A: Why did you make this conclusion?
• Q: Because the Lorentz transformation of the field ψ(x,t) involves intermixing of the arguments x and t, which is only possible if x and t are eigenvalues of operators X and T that do not commute with the boost generator.
• A: I guess you may be right.
• Q: So, there is no connection between quantum field and quantum-mechanical wavefunctions. And quantum fields are just abstract operator functions on an abstract 4-dimensional space whose coordinates are not necessarily related to experimentally observed positions and times.
• A: Why do you call Minkowski space-time "abstract"?
• Q: We will talk about Minkowski space-time later. However, arguments x and t of quantum fields are just dummy integration variables without any particular physical meaning.
• A: Why it is so?
• Q: Because, as we discussed here, the only physical thing predicted by relativistic QFT is S-matrix. And in the expression for the S-matrix quantum fields enter integrated over x and t. All dependence on x and t is lost.

P.S. Everything said above refers only to relativistic quantum fields used in theories like QED. They should be separated from non-relativistic quantum fields used, for example, in condensed matter physics. The quantum field of phonons is a well-defined operator which is related to displacements of real atoms from lattice positions.

How we can improve QED?

Let us now count arguments for and against the currently accepted form of the renormalized QED. We start with positive things:

1. QED is relativistically invariant
2. QED is cluster separable
3. The S-matrix of QED perfectly agrees with experiment.

Now the negatives:

1. The Hamiltonian of QED contains infinite counterterms
2. QED interaction leads to unphysical processes of creation and absorption of extra particles from vacuum and 1-particle states
   

Is it possible to reformulate the theory so as to keep the positive stuff and get rid of the negative? Yes. The answer is here. The Hamiltonian of QED is apparently wrong, but according to Ekstein, we can modify the Hamiltonian without changing the S-matrix (i.e., without changing the agreement with experiment). This can be done by a unitary transformation. Moreover, if we are careful enough, we can choose this transformation such that both the relativistic invariance and the cluster separability are not affected. Now, the question is, whether one can simultaneously eliminate bad properties 1. and 2? And the answer is, again, 'yes'! One can find a unitary transformation of the Hamiltonian such that all interaction terms become finite, and all terms that act non-trivially on the vacuum and 1-particle states are eliminated. In one shot we kill two rabbits: with the new Hamiltonian the renormalization is not needed anymore and we have a clean physically satisfactory definitions of the vacuum and 1-particle states. This is called the "unitary dressing transformation". More about it in another post.

Update So, the conclusion is that the QED Hamiltonian is simply wrong and must be substituted by another Hamiltonian. Why did we use a wrong Hamiltonian for almost 80 years? One reason is in three "good" properties I listed above. This is a respectable reason. Another reason is that the Hamiltonian of QED was derived from correspondence with classical Maxwell's theory of electromagnetism. This is, actually, not a good reason. As we discussed here , Maxwellian description of radiation by a continuous wave is not accurate. There are numerous other flaws in Maxwell's theory, one of them being inadequate description of the "radiation reaction" force. So, derivation of quantum electrodynamics by "quantizing" classical theory is, at best, a heuristic trick.

What's wrong with the Hamiltonian of QED?

I mentioned several times on this blog that the non-trivial action of the QED interaction V on the vacuum and one-particle states is deeply disturbing. What's wrong with it?

The interaction Hamiltonian V written in terms of particle creation and annihilation operators contains so-called tri-linear terms. An example of such a term is the product of three particle operators a*c*a, where I denoted 'a*' an electron creation operator, 'a' an electron annihilation operator, and 'c*' a photon creation operator. Let us focus just on this term and write the full Hamiltonian H in the truncated form (all momentum, spin, and polarization labels are omitted as well as numerical factors, integration signs, etc.)

H = H_0 + a*c*a + ... (1)

Having the full Hamiltonian H one can form the time evolution operator U = exp(iHt) and study the time dependence of states and observables. Let us consider the time evolution of a state a*|0> which at time t=0 had only one electron (here |0> denotes the vacuum state). We would expect (according to observations) that a single electron always remain in the one-electron state. Let us now find out whether this expectation is fulfilled with our Hamiltonian. We are going to find out how this state will look like after short time t. Let us take t small enough, so that the time evolution operator U can be approximated by linear terms in the series over t. Then

exp(iHt) a*|0> = exp(it(H_0 + a*c*a + ...)) a*|0>
= (1 + it H_0 + it a*c*a + ...) a*|0>
= a*|0> +it H_0 a*|0> + it a*c*a a*|0> + ... (2)

The first two terms in this expansion are harmless. They both correspond to states with one electron, as expected. However, the third state is different. After bringing the particle operators to the normal order and omitting, again, all numerical factors, this term will look like

a*c*|0> (3)

and this is now a state with one electron and one photon. So, the time evolution with the QED Hamiltonian (1) leads to the "emission" of photons by a single free electron. Such an effect has never been seen experimentally. The problem becomes even more severe if we take more terms in the expansion of the operator U and consider additional terms in the interaction V. It is easy to show that the vacuum state |0> will also become unstable with respect to the decay into multiparticle states containing photons and electron-positron pairs. This is also unphysical.

The cure for this desease is known for many years. It is often said that creation and annihilation operators, like a* and a, do not correspond to real physical particles. They describe creation and annihilation of fictitious "bare" particles. Real electrons are complex linear combinations of multiparticle "bare" states. Although, this has been known for decades, there were suprisingly few attempts to reformulate QED in terms of physical particles that can be observed. In view of the complete failure of QED to describe the time evolution, such reformulation of the theory seems to be a reasonable step. This step is exactly the essence of the "dressed article" program initiated in 1958 by the work of Greenberg and Schweber [1].

I would like to mention just one more thing. It appears that in a properly dressed theory all ultraviolet divergences disappear from both the S-matrix and the Hamiltonian. More about the "dressed particle" approach in later posts.

[1] O. W. Greenberg and S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim., 8, (1958) 378.

How we can improve QFT?

Here we discussed that QFT in its current form cannot describe the time evolution of wave functions and observables. Here we quoted Weinberg as saying that QFT is the only way to unify relativity with cluster separability and that any fundamental theory in the low-energy limit should look like QFT. So, what shall we do now? Give up? No.

It appears that Weinberg's profesy is wrong. There is a way to construct a low-energy fundamental relativistic quantum theory that does not look like QFT. Examples of such theories were constructed by Kita in [1] and in several later papers. Similar ideas were explored by Kazes [2]. By following these approaches, one can define interaction operators in H = H_0 + V and K = K_0 + W (see here ) so that (i) the Poincare commutators are satisfied, (ii) the interaction operators are cluster separable, (iii) V and W yield zero when acting on the vacuum and 1-particle states. The last condition is important to have a well-defined time evolution in the theory.

The bad news about Kita's theory is that he was able to construct only simple models that apparently have no counterparts in nature. The good news is that his approach can be applied to realistic theories, such as QED. A satisfactory version of QED with properties (i), (ii), and (iii) can be constructed by using the "dressed particle" approach. More about that in another posting.



[1] H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties", Progr. Theor. Phys., 35, (1966) 934.

[2] E. Kazes, "Analytic theory of relativistic interactions", Phys. Rev. D, 4, (1971) 999.

Is there an observable of time?

We already discussed here, here, and here that the principle of relativity does not require the symmetry between space and time coordinates alleged in Einstein's special relativity. Here I would like to suggest another argument for the deep division between time and position. The point is that position is an observable, while time is not.

When we measure position of a particle or any other physical system, we know that the result of measurement will depend on the state of the observed system.
In quantum mechanics there is an Hermitian operator corresponding to position, and observations of the position are normally associated with uncertainties that are characteristic to measurements of quantum observables.

Time is different. Time is "measured" by simply looking at the wall clock in the laboratory. The value of time does not depend on what physical system is actually observed and what is the state of the system. The value of time would be just the same if we haven't observed any physical system at all. So, time is simply a classical parameter associated with the act of measurement. We perform a measurement of some observable and we attach to this measurement a numerical label called "time". This label belongs to the same class as nine other labels that should be attached to each measurement. These are labels that uniquely specify the reference frame in which the measurement was made: its position, orientation, and velocity. So, time is just one of the 10 parameters of the reference frame or laboratory, and time has nothing to do with the properties of the physical system that may be (or may be not) present in the laboratory.

So, it is not surprising that there is no operator of time in quantum mechanics. (Some published attempts to introduce such an operator do not look convincing.) This observation is just another reason to stop using the Minkowski space-time formalism in relativistic theories.

What is observable and what is not?

In physics, there are some things that can be directly observed by experiment, and there are other things that exist only in theories and are not observable. The things of the first kind are, for example, various properties of particles, such as mass, spin, momentum, position, etc. There are lot of experimental devices which measure these properties directly: beginning from simple rulers and ending with Wilson chambers and Stern-Gerlach apparatuses. Examples of things of the second kind (non-observable) are fields and space-time.

Take for example, the electromagnetic field. There is no way one can directly measure the strength of the field (electric or magnetic) at a given point. All we can do is to place a test charge at this point and measure the force acting on this charge. One is free to think that this force appears because of the non-zero field vector created at this point by other charges. However, one can also think that there is no field and the force is simply created by action-at-a-distance from the surrounding charges. In classical electrodynamics, one also assigns certain momentum (density) and energy (density) to the fields. In the case of static fields this momentum-energy is certainly non-observable. In the case of freely propagating transversal field (= light wave) the momentum-energy can be equally well assigned to the particles of light - photons (see also discussion here ). So, the idea of electromagnetic field as a separate physical entity is somewhat suspicious.

Now consider the space-time. In modern theories the space-time is an active participant in physical processes. It can be bent, twisted, torn... It even has momentum-energy associated with it (with the "gravitational field"). However, nobody can see the space-time properties directly. What we actually see in experiments are trajectories of particles, i.e., the time dependent expectation values of the position observable r(t). These trajectories can be calculated in the Hilbert space formalism of quantum mechanics, where the notion of space-time is just not needed.

I believe that a successful physical theory should be formulated (as much as possible) in terms of directly observable quantities (= particle properties described above). I strongly believe that current crisis in theoretical physics in large part is related to our focus on abstract theoretical non-observable notions, such as fields and space-time.

Can relativistic QFT predict time evolution?

From what I know about relativistic quantum field theory, the answer is "no" (if you disagree, please provide a reference that proves otherwise). The explanation is simple. In quantum mechanics, one needs to have a well-defined Hamiltonian in order to form the time evolution operator exp(iHt) and to calculate the time dependence of wave functions and observables. However, in relativistic QFT, the only sensible Hamiltonians (i.e., those that can be used to calculate the S-matrix via Feynman-Dyson formula) must contain infinite renormalization counterterms. So, they are not well-defined. They are actually not defined at all. Even if we forget about the infinities in QFT Hamiltonians (for example, we can introduce artificial momentum cutoffs), they are still not good, because they normally contain terms (like trilinear terms in QED) which have a non-trivial action on the vacuum and one-particle states. This is completely at odds with the observed stability of vacuum and single particles.

QFT is good at calculating one and only one thing - the S-matrix (or the S-operator). All wonderful experimental predictions of QED or Standard Model (scattering amplitudes, anomalous magnetic moments, Lamb shifts, etc.) are related to the S-matrix elements. The time evolution enters in the S-matrix in an integrated form (from infinite past to infinite future). It is just impossible to recover the detailed form of a function by knowing its definite integral. By having exact knowledge about the S-operator we can say very little about the underlying Hamiltonian. It has been shown [1] that there exists a huge class of scattering-equivalent Hamiltonians connected to each other by unitary transformations.

So, the great successes of relativistic renormalized QFT in calculations of the S-matrix and related observable quantities do not at all contradict the miserable performance of QFT when it comes to calculations of the time evolution. This fact went unnoticed for a long time for a simple reason. Currently there are no experimental techniques capable of measuring the detailed time evolution of micro-particles. Existing time-dependent experimental data are of such a low quality that simple non-relativistic Hamiltonians are quite capable to describe them, and fine relativistic and radiative corrections are just not needed.

There is a way to fix this problem, i.e., to have a finite well-defined Hamiltonian which is useful for both S-matrix and time evolution calculations in QFT. This way is called the dressed particle approach. More about that in another posting.

[1] H. Ekstein, "Equivalent Hamiltonians in scattering theory", Phys. Rev. 117 (1960), 519

What is the meaning of relativistic invariance?

It is usually assumed that in order to be relativistically invariant a physical theory must be formulated in terms of manifestly covariant quantities, i.e., quantities that transform as scalars, vectors, tensors, etc. with respect to boost transformations. For example, relativistic quantum theories of single particles were initially formulated by using wave equations (Klein-Gordon, Dirac, etc.) in which the time and position variables were formally equivalent. We already discussed that the tensor transformation laws of observables and the formal equivalence of the time and position coordinates are approximations. Then what is the exact meaning of relativistic invariance? How one can say whether a theory is relativistically invariant or not?

The key to understanding the relativistic invariance is not in the formal equivalence of space and time coordinates. The key is in the (Poincare) group properties of the ten types of inertial transformations (1 time translation, 3 space translations, 3 rotations, and 3 boosts). A quantum theory if fully relativistic if there is an unitary representation of the Poincare group in the Hilbert space of the system, and if the generators H for time translations, P for space translations and J for spatial rotations of this representation are identified with the observables of the total energy, total momentum and total angular momentum, repsectively, of the system. A relativistic classical theory is defined similarly. The difference is that instead of the "Hilbert space" one should use the "phase space", and instead of "unitary transformations" one should use "contact transformations", i.e., those conserving the Poisson brackets.

Is interference a quantum effect?

Consider the famous double-slit experiment with photons. This experiment can be performed in two regimes: the low intensity (Feynman) regime and the high intensity (Young) regime. In the Feynman regime, photons are released one-by-one, the image on the screen is built one dot at a time, and the explanation of the interference picture is given by quantum mechanics of particles - photons. In the Young regime, the interference picture is exactly the same as in the Feynman regime (after enough dots were accumulated on the screen, so that a continuous distribution of the intensity emerged), however, the explanation of the interference is radically different. Classical electrodynamics describes light as a continuous electromagnetic field. One interference picture, two different explanations. Which one is correct?

Note that there is no any significant difference between these two regimes apart from the intensity of light or the number of emitted particles. Therefore it would be not correct to say that the Young regime arises in some kind of classical limit, i.e., when the Planck constant can be neglected. The same physical mechanism works in both regimes, and the theoretical explanation should be also the same. The only explanation that works in both cases is the view that light is a flow of particles - photons, and that the wave properties of light are manifestations of the quantum nature of these particles.

This brings up interesting questions. Are we making a mistake when calling Maxwell's wave theory of light a "classical theory"? Is wave theory of light, in fact, a surrogate attempt to describe quantum effects? I think the answers to both these questions should be "yes". Newton's rings, Grimaldi's diffraction, and Young's interference are genuine quantum effects, and their correct description requires the machinery of quantum mechanics: Hilbert spaces, wave functions, etc.

Newtonian ray optics in which light corpuscles move along well-defined trajectories is the only truly classical description of light.