Time evolution in quantum field theories

[A. Neumaier]

On a second thought I've decided that I'm not satisfied with this answer. The confining potential and the external field are supposed to be independent on the number of particles in the dot.

I derived everything form a more fundamental model, and nothing in this derivation suggests that any of the terms you want to have vanish should be absent. If you want to insist on your view, please tell me which part of my derivation is faulty. The derivation was elementary, so it should be easy to spot an error in it.

 

[A. Neumaier]

This is a big difference between out philosophies. I don't think that "particles are constructed from the Hamiltonian". In my opinion, particles are given to us a priori. The Hamiltonian is an operator, which we write down to describe the interaction between particles.

I didn't say ''particles'' are constructed, but ''operators'' are constructed. Of course, the particles are given.

I was probably not very clear on this point before, but I would like to stress it here. I don't consider "dressing transformation" as a desirable way to perform calculations in QFT.

In the present case you don't need any dressing transformation since everything is manifestly finite and as well-defined as the underlying nonrelativistic dynamics with H_full from which the simplified model was derived. So there is no reason not to solve the exercise.

I mentioned dressing only because you wanted to get rid of the interaction terms with less than two annihilators or creators. This is possible without changing the validity of the model only if you do it via a unitary transform, i.e., using a dressing transformation.
But I agree it is an undesirable way of doing the calculations. And the original exercise doesn't need such a detour - you'll encounter nothing unphysical.

The desirable way is to define the Hamiltonian, so that there are no unphysical self-interactions

In the present case, the self-interactions are not unphysical but generated by the projection to the main degrees of freedom, which simplifies a complex space-time problem to a simple quantum dot.

With this good Hamiltonian there can be no difference between "bare" and "physical" particles. I believe that this is the only appropriate form of the Hamiltonian in QFT.

You _want_ that, but the derivation proves that one gets something different.

Unfortunately, Hamiltonians in existing QFT theories do not obey this principle.

No effective Hamiltonians in solid state physics obeys this principle. It is not appropriate for this kind of problems.

Now you suggest to consider a theory, which is formulated in this inappropriate non-transparent self-interacting way from the beginning.

No. I suggest to consider a toy problem derived a well-defined microscopic nonrelativistic Hamiltonian for dot+particles that satisfies your requirement. I motivated the reduced Hamiltonian by explicitly deriving everything from the underlying full theory. Nothing in this derivation is inappropriate, and it is fully transparent.

So, you invite me to do the cleanup myself. Yes, I can do that following the procedure outlined in the book. Then I would obtain a well-defined Hamiltonian H for physical particles without self-interactions.

You don't need the cleanup if you use instead my well-defined Hamiltonian H for physical particles with self-interactions. Your dressing would just replace my physical particles (which are identical with the microscopic particles) by effective particles.

I believe that this is the true Hamiltonian, which can be used in routine quantum mechanical calculations without any tricks and renormalizations.

You don't need any tricks. The resulting renormalizations are precisely the same energy shifts that you'd get when you'd solve the anharmonic oscillator by perturbation theory.

For example, if we diagonalize the Hamiltonian H we obtain energies and wave functions of stationary states, that can be compared with experiments. If |\psi(0> is an initial state vector, then exp(iHt)|\psi(0> is the state vector evolved to time t. This is how I understand the title of this thread "Time evolution is quantum field theories".

One gets exactly the same dynamics, whether one works in the representation with the free particles or in the representation with the dressed particles. There is no more difference than the difference between working in the position or the momentum representation.

Now you are saying that my understanding is wrong and there should be a different approach to the time evolution - the one based on Wightman functions. In order to "build my intuition" you suggest to immerse into all these calculations with self-interacting "bare" particles, non-trivial vacuum, renormalization, etc. These calculations are meaningless, in my opinion.

You call it meaningless - against a long and successful tradition of using it. I could teach you how to assign meaning to what you consider meaningless. But only if you do the exercise. With the effort you spent in discussing all that you'd have already solved it, and we could progress...

 

[meopemuk]

I derived everything form a more fundamental model, and nothing in this derivation suggests that any of the terms you want to have vanish should be absent. If you want to insist on your view, please tell me which part of my derivation is faulty. The derivation was elementary, so it should be easy to spot an error in it.

I'm afraid that the original model has been modified already, e.g., by introduction of the external field. If it is not difficult for you, could you please formulate the physical model again? As I understand, this model includes (1) the confining potential, (2) N particles inside the potential well, (3) external field. So, I would like to know what interacts with what in this model, and what are the physical assumptions. I am most interested in interactions that are responsible for the q^3+q^4 term in the Hamiltonian, especially what is the mechanism by which 0-particle and 1-particle states are no longer eigenstates of the full Hamiltonian?

Eugene.

 

[A. Neumaier]

I'm afraid that the original model has been modified already, e.g., by introduction of the external field. If it is not difficult for you, could you please formulate the physical model again? As I understand, this model includes (1) the confining potential, (2) N particles inside the potential well, (3) external field. So, I would like to know what interacts with what in this model, and what are the physical assumptions.

On a very detailed level, the system consisting of the quantum dot and the bosons are described in the absence of the external field (g=0) by a Hamiltonian H_full^0. The external field with potential g A(x) causes an interaction V_full(g), obtained by integrating the product of g A(x) with the expression for the current operator. Thus V_full(g)=g H_full^1, leading to the total Hamiltonian H_full(g) = H_full^0+V_full(g) = H_full^0+g H_full^1. Post #60 (which uses some background notation from post #58) shows how to get from there to the effective potential in a Fock space where particles have only one degree of freedom.

Since we do not know the microscopic description, the resulting matrix elements V_jk can be anything - there is nothing in the microscopic model that would force any of the coefficients to be small, while we can assume that high order contributions can be neglected since they only matter for high energy excitations but we assume that g is tiny and we start with a few-particle initial state.

In a real quantum dot calculation we therefore would have to take all V_jk with j+k<=4, say as parameters, solve the problem, gather experimental data and do a least squares fit to the results of the theoretical prediction in order to get an estimate for the actual V_jk. Again, there is nothing in the experimental record that would force any of these coefficients to be small, unless there are many accidental cancellations.

I am most interested in interactions that are responsible for the q^3+q^4 term in the Hamiltonian, especially what is the mechanism by which 0-particle and 1-particle states are no longer eigenstates of the full Hamiltonian?

Probably there is nothing in a real quantum dot that would fix the interaction values to precisely those values corresponding to a quartic potential in q. Instead, one would get an arbitrary quartic interaction potential V(p,q). If you insist on being fully realistic, you'd have to do this general case. It is obvious that this interaction neither preserves the 0-particle state nor the 1-particle state, unless there are many accidental cancellations.

However, in order to keep the work reasonably low, and because the real purpose of the exercise is to get insight into the Wightman representation for Lagrangian field theories (rather than tuning an experimental quantum dot), we fix the quartic interaction potential to the simple form V(p,q)=aq^3/3+bq^4/4, corresponding to a Lagrangian Phi^4_1 field theory, with
H(g)=(p^2+q^2)/2 + g (aq^3/3+bq^4/4).
And as mentioned before, it suffices to treat either the symmetric case a=0, b=1 or the unphysical but simpler limiting case a=1, b=0. This is enough to get the necessary insight, and achieves it with a minimum of calculations.

 

[meopemuk]

Probably there is nothing in a real quantum dot that would fix the interaction values to precisely those values corresponding to a quartic potential in q. Instead, one would get an arbitrary quartic interaction potential V(p,q). If you insist on being fully realistic, you'd have to do this general case. It is obvious that this interaction neither preserves the 0-particle state nor the 1-particle state, unless there are many accidental cancellations.

However, in order to keep the work reasonably low, and because the real purpose of the exercise is to get insight into the Wightman representation for Lagrangian field theories (rather than tuning an experimental quantum dot), we fix the quartic interaction potential to the simple form V(p,q)=aq^3/3+bq^4/4, corresponding to a Lagrangian Phi^4_1 field theory, with
H(g)=(p^2+q^2)/2 + g (aq^3/3+bq^4/4).
And as mentioned before, it suffices to treat either the symmetric case a=0, b=1 or the unphysical but simpler limiting case a=1, b=0. This is enough to get the necessary insight, and achieves it with a minimum of calculations.

I can always place one particle in the real quantum dot. This state is an eigenstate of the total Hamiltonian. In your model the 1-particle state a*|0> is not an eigenstate of the total Hamiltonian, due to the presence of q^3 and/or q^4 interaction terms. I can see only two explanations for this discrepancy:

(1) Your Hamiltonian is wrong

(2) a and a* are not a/c operators of real particles.

Which explanation is the correct one?

Eugene.

 

[A. Neumaier]

I can always place one particle in the real quantum dot. This state is an eigenstate of the total Hamiltonian.

This state is an eigenstate of the total Hamiltonian when g=0. But when the interaction is switched on it, it is no longer an eigenstate, except in the case where it is accidentally also an eigenstate of V. But there is no physical reason why this accident should happen.

The same happens already for the empty dot. It is obvious that a physical system (and the dot is such a system) changes its ground state when an interaction is switched on. Thus the vacuum state (and the once occupied state) changes as a function of g.

In your model the 1-particle state a*|0> is not an eigenstate of the total Hamiltonian, due to the presence of q^3 and/or q^4 interaction terms. I can see only two explanations for this discrepancy:

(1) Your Hamiltonian is wrong

(2) a and a* are not a/c operators of real particles.

Which explanation is the correct one?

You forgot the third, correct explanation: Your intuition from QED where it is impossible to switch off interactions is inappropriate for the interpretation of problems in a controllable external field.

 

[meopemuk]

This state is an eigenstate of the total Hamiltonian when g=0. But when the interaction is switched on it, it is no longer an eigenstate, except in the case where it is accidentally also an eigenstate of V. But there is no physical reason why this accident should happen.

The same happens already for the empty dot. It is obvious that a physical system (and the dot is such a system) changes its ground state when an interaction is switched on. Thus the vacuum state (and the once occupied state) changes as a function of g.

If I've placed one particle in the dot it stays there as one particle no matter whether the external field is off or on. The wave function of the particle can change depending on the field, but the number of particles (=1) does not change. In your suggested model the external field has the capability to affect the number of particles inside the dot, which is very unusual, to say the least.

Eugene.

 

[A. Neumaier]

If I've placed one particle in the dot it stays there as one particle no matter whether the external field is off or on. The wave function of the particle can change depending on the field, but the number of particles (=1) does not change. In your suggested model the external field has the capability to affect the number of particles inside the dot, which is very unusual, to say the least.

If you look at the construction of the model you'll find that the ''particles'' are not the particles in the absence of the dot but already effective particles whose state is defined through the environment given by the dot in the absence of the external field. In reality, i.e., seen from a microscopic point of view, these are superpositions of the isolated particle and contributions from the dot.

Only the total number of particles comprising the dot, its content, and the whole surrounding supporting it is conserved - not the projection of the particle number to the area of the dot. This is enough to account for changes in particle number when the potential is switched on. It is enough that the external field generates a nonzero probability for the particle to be outside the region of the dot chosen to create the reduced model - but not far enough that the particle can leave the dot.

There is no way to avoid that if the electric field is at all capable of changing the number of particles in the box. If this is not possible, we have no quantum dot but a permanently bound particle.

 

[meopemuk]

...the ''particles'' are not the particles in the absence of the dot but already effective particles...

OK, my confusion has reached a dangerous level. I am bailing out. Thank you for your time and effort.

I would appreciate if you can recommend books/articles where the idea of the "Wightman Hamiltonian" and time evolution in QFT is explained at the most basic level. I will study it at my own pace.

Thanks.
Eugene.

 

[A. Neumaier]

I would appreciate if you can recommend books/articles where the idea of the "Wightman Hamiltonian" and time evolution in QFT is explained at the most basic level. I will study it at my own pace.

People working in QFT proper don't usually think in terms of Hamiltonians, except for the very technical papers that prove correlation bounds to prove the existence of a quantum field theory. Hence they do not explain these things - I had to find out everything by reading between the lines. And people doing kinetic or hydrodynamic studies simply use the CTP formalism based upon functional integral approach to the Wightman functions rather than canonical quantization. There the Hamiltonian issue doesn't arise at all, and the dynamical interpretation is obvious since the resulting equations look like classical effective field equations with quantum corrections.

Thus I can't point you to anything useful. If my simple exercises (specifically created to match your interests) were already too much for you, you'll hardly have the perseverance to read through algebraic quantum field texts. I had given you some entry points to closed time path (CTP) methods, but none of these explain the relations to a particle picture.