Time evolution in quantum field theories

[meopemuk]

In d+1 dimensions with d>0, your form (2) is ruled out by Lorentz invariance; to maintain Lorentz invariance, interactions must be written in terms of q. See Weinberg vol.I or chapter 4 of Srednicki.

No. There is no proof that with the Hamiltonian (2) you cannot build a relativistically invariant theory. It is true that existing relativistic quantum field theories do not use form (2), but neither Weinberg nor Srednicki nor somebody else has proven that there is no other way. There are actually counterexamples in which the Hamiltonian of the form (2) satisfies approximately the necessary commutation relations, and there are good reasons to believe that it is possible to satisfy these relations exactly (i.e., in all orders of perturbation theory). For mode discussions of this issue you can visit an Independent Research thread https://www.physicsforums.com/showthread.php?t=474666 [/URL].

Eugene.

 

[A. Neumaier]

In d+1 dimensions with d>0, your form (2) is ruled out by Lorentz invariance; to maintain Lorentz invariance, interactions must be written in terms of q. See Weinberg vol.I or chapter 4 of Srednicki.

We are discussing here a quantum dot, which is 1+0D QFT. The 1-dimensional Lorentz group is trivial, hence Lorentz invariance poses no restriction.

 

[A. Neumaier]

The next logical step would be to allow the particles in the box to interact. Interaction takes at least two particles to participate. So, addition of interaction cannot have any effect on zero-particle and one-particle states and energies. Only 2-particle, 3-particle, etc. states and energies can be affected by interaction. From this condition it is clear that any reasonable interaction that can be added to the free Hamiltonian (1) must have the form (I drop numerical coefficients in front of operator symbols as I am interested only in the general operator structure of the terms)

V = a^{\dag} a^{\dag}aa + a^{\dag}a^{\dag}aaa + a^{\dag}a^{\dag}a^{\dag}aa + \ldots...(2)

The characteristic feature of this operator is that in the normally-ordered form it has at least two annihilation operators and at least two creation operators in each term. The ellipsis at the end indicates that more complex terms with these features can be added there.

Now, you are suggesting something completely different. Your interaction V= aq^3/3+bq^4/4 being expanded in a^{\dag}, a, does not have the form (2). This means that 0-particle and 1-particle states and energies are affected by your interaction. This means that interaction has changed the definition of particles. Your new 1-particle state (which can be defined as the state with the 2nd lowest total energy value) is a linear combination of eigenstates of the old H. So, by introducing interaction V= aq^3/3+bq^4/4 you have changed the physics of your quantum dot in a very dramatic way. Your new physical vacuum is different from the old (bare) vacuum. Your new physical particles are different from the old (bare) particles. All your theory is formulated in terms of bare (=meaningless) operators a^{\dag}, a which do not correspond to any physical thing anymore. You have created a lot of problems by introducing a completely unphysical interaction operator. These problems will lead you to the need of doing renormalizations and other headaches down the road.

That's precisely one of the points of the exercises - you'll learn to understand the meaning of renormalization in a simple case where there are no divergences. Everything is harmless. The point is that in solid state physics (and hence in a quantum dot), interactions turn free particles into effective particles that are _different_ from the original ones. (For example, effective photons in glass are slower than free photons in vacuum.)

But in the present context you'd use the term ''free'' in place of bare, since the interaction can be switched on and offf (by changing the confining magnetic fields); so they have a physical interpretation (quite unlike the case in relativistic QFT, where switching off the interactions is impossible).

I am discussing the anharmonic oscillator in the Fock representation, or, equivalently, the quantum dot, a 1+0-dimensional QFT defined by a quartic Lagrangian. Both are well-defined quantum systems, and the fact that one has two different interpretations of the same abstract model is a big advantage, which I intend to exploit didactically.

There the interaction is given by a quartic polynomial, expanded into c/a operators. This produces interaction terms that are not only of the form you want, but are at most quartic. These interaction terms are physically important, as they generate
(i) in the anharmonic oscillators the physically measurable line shifts if an interaction is turned on;
(ii) in the quantum dot the renormalization of the vacuum and the particle mass. In 1+0D, these effects are finite, hence mathematically and physically respectable. And by doing the exercises you'll learn to interpret these things correctly.

 

[A. Neumaier]

This is completely wrong.

I replied in the thread https://www.physicsforums.com/showthread.php?p=3185296

 

[meopemuk]

There the interaction is given by a quartic polynomial, expanded into c/a operators. This produces interaction terms that are not only of the form you want, but are at most quartic. These interaction terms are physically important, as they generate
[...]
(ii) in the quantum dot the renormalization of the vacuum and the particle mass. In 1+0D, these effects are finite, hence mathematically and physically respectable.

I don't see anything "physically respectable" in the renormalization of the vacuum and the particle mass. No matter how strong is interaction between two or more particles, this cannot have any effect on a single particle (which has nothing to interact with). There can be no effect on the vacuum at all, because there are no particles and no interactions in the vacuum, by definition.

In your example you've introduced a Hamiltonian, which is mathematically simple (quartic), but physically unacceptable. This Hamiltonian violates the important rule that interaction can affect only states with two or more particles. So, you've created a huge problem for yourself, which you are then going to solve with "heroic" renormalization efforts.

Why do you find it necessary to define interactions so that the vacuum and 1-particle states are affected? Why don't you like my proposal with interaction a*a*aa, which would make everything much simpler?

Eugene.

 

[meopemuk]

The point is that in solid state physics (and hence in a quantum dot), interactions turn free particles into effective particles that are _different_ from the original ones. (For example, effective photons in glass are slower than free photons in vacuum.)

I don't find this reference to solid state physics convincing. In solid state physics you have a "medium", e.g., a crystal. In the quasiparticle picture you don't treat this medium explicitly, but model its presence implicitly by renormalizations. If I am interested in an electron moving through empty space, there is no "medium" of any kind. Likewise, there is no "medium", when I am studying particles placed in a small box. These two cases should not involve renormalizations.

Eugene.

 

[A. Neumaier]

I don't find this reference to solid state physics convincing. In solid state physics you have a "medium", e.g., a crystal. In the quasiparticle picture you don't treat this medium explicitly, but model its presence implicitly by renormalizations.

A quantum dot cannot exist except in a medium that constrains the system to a dot. Thus the medium is essential. In the approximation considered here, the medium is represented by the nonquadratic terms in the potential. This causes the renormalizations. The renormalizations are needed - otherwise we are no longer solving the original Lagrangian system.

If I am interested in an electron moving through empty space, there is no "medium" of any kind. Likewise, there is no "medium", when I am studying particles placed in a small box. These two cases should not involve renormalizations.

First things first. One cannot understand the quantum field theory of an electron on a deeper level without first having understood the quantum dot.

We can do the electron later, after having progressed from the quantum dot to nonrelativistic solid state physics and then to relativistic field theory.

 

[A. Neumaier]

I don't see anything "physically respectable" in the renormalization of the vacuum and the particle mass. No matter how strong is interaction between two or more particles, this cannot have any effect on a single particle (which has nothing to interact with). There can be no effect on the vacuum at all, because there are no particles and no interactions in the vacuum, by definition.

Our quantum dot is a solid state device with a confining potential, defined by a quartic Lagrangian (and hence the Hamiltonian I gave), which has an interaction that can be tuned by changing the factor g in front of a and b. The ''vacuum'' is the ground state of this interacting system. Thus , in a quantum dot, the vacuum is simply the unoccupied dot - far from the vacuum you imagine. The free vacuum is the empty dot in which the confining potential is harmonic (g=0). The dressed vacuum is the emptied dot in which a perturbation is switched on; of course this changes ground state and hence renormalizes the vacuum.

In your example you've introduced a Hamiltonian, which is mathematically simple (quartic), but physically unacceptable.

A real quantum dot can probably have a fairly arbitrary function of a and a^* as the Hamiltonian; certainly there is nothing that would forbid the terms that you don't want to have. They are essential for modeling the change in the ground state energy of the combined system (quantum dot + confining matrix) when the interaction is switched on.
If the quantum dot is formed by arbitrarily many bosons with only two energetically accessible states psi_0 (unexcited) and psi_1 (excited) - both functions of 3-position or 3-momentum -, the state space is the space spanned by the states |k> consisting of the symmetrized tensor product of k bosons in state psi_1 and all other bosons in state psi_0. Thus the N-particle state is the state of N excited bosons, and a ''particle'' is simply an excitation. The effective quantum dot Hamiltonianl is
H = sum H_jk (a^*)^ja^k,
where H_jk=<j|H_full|k> is computed by taking matrix elements of the full many-particle Hamiltonian in the Fock space over L^2(R^3). There is no reason why some of the H_jk should be zero. Thus a general quantum dot has no restrictions on the form of the potential.

Why do you find it necessary to define interactions so that the vacuum and 1-particle states are affected? Why don't you like my proposal with interaction a*a*aa, which would make everything much simpler?

The main purpose of the exercise is not to do computations for a realistic quantum dot, but to have illustration material for the Wightman representation. The latter is a representation for field theories defined by Lagrangians that are at most quadratic in the derivative of the fields.

In order to serve our purpose, our quantum dot must therefore correspond to a Lagrangian L(q)= m qdot^2/2 -V(q). (One could also add first order terms in qdot; but this makes things only more complicated.) Doing the usual Legendre transform results in H=p^2/2m+V(q).

If you want to have things as simple as possible, take V(q) = k q^2/2+g q^3/3. While this is unphysical, it is the limiting case c->0 of the quartic potential V(q) = k q^2/2+g (q^3/3+c q^4/4) (which has unbroken symmetry for 0<g<4kc), and has therefore a fully adequate perturbation theory (where g is infinitesimally small). Note that some wellknown textbooks on renormalization start for the same reason with the unphysical phi^3 theory!

 

[meopemuk]

A quantum dot cannot exist except in a medium that constrains the system to a dot. Thus the medium is essential. In the approximation considered here, the medium is represented by the nonquadratic terms in the potential. This causes the renormalizations. The renormalizations are needed - otherwise we are no longer solving the original Lagrangian system.

Please resolve my confusion. I am still not sure about the formulation of this problem. Assume that the "quantum dot" is prepared as a vacancy in semiconductor or something of that sort. We allowed to put one or more bosons into this vacancy. The vacuum in this case corresponds to the empty vacancy. So, the energy of this "vacuum" is the total energy of the crystal with empty vacancy. Now we start to put bosons into the vacancy one-by-one. Of course, the bosons interact with the crystal (through the confining potential of the vacancy), but they don't interact between themselves. So, with each new boson the total energy of the system increases by fixed energy E. This is expressed by writing the model Hamiltonian as

H = Ea*a = p^2/2m + kq^2/2...(1)

My question is: where the non-quadratic terms in the Hamiltonian come from? I thought that these terms were supposed to express the (originally neglected) mutual interaction between bosons. Then these additional terms cannot have any effect on the 0-boson (=empty vacancy) and 1-boson (vacancy+one boson) states. So, these terms *cannot* have the structure ~q^3 + q^4. I propose interaction ~a*a*aa as a reasonable alternative. If the non-quadratic terms are some corrections to the boson-crystal interactions, then why didn't we took them into account in our original formulation of the Hamiltonian (1)?

Eugene.

 

[A. Neumaier]

Assume that the "quantum dot" is prepared as a vacancy in semiconductor or something of that sort. We allowed to put one or more bosons into this vacancy. The vacuum in this case corresponds to the empty vacancy. So, the energy of this "vacuum" is the total energy of the crystal with empty vacancy. Now we start to put bosons into the vacancy one-by-one. Of course, the bosons interact with the crystal (through the confining potential of the vacancy), but they don't interact between themselves. So, with each new boson the total energy of the system increases by fixed energy E. This is expressed by writing the model Hamiltonian as

H = Ea*a = p^2/2m + kq^2/2...(1)

My question is: where the non-quadratic terms in the Hamiltonian come from? I thought that these terms were supposed to express the (originally neglected) mutual interaction between bosons.

No. As always in perturbation theory in QM, the Hamiltonian is assumed to be parameterized by a parameter g, denoting for example the strength of an external electric or magnetic field. Thus
<br /> H(g)= \sum_{j,k} H_{jk}(g) (a^*)^ja^k,~~~ H_{jk}(g)=&lt;j|H_{full}(g)|k&gt;.<br />
Since the e/m field interacts linearly, H_jk(g) is linear in g, hence H=H(0)+gV where V has (the in general arbitrary) matrix elements V_jk=dH_jk(g)/dg. On the other hand, H(0) - the Hamiltonian when the e/m field is switched of f - was used to construct the basis for the reduced description, hence has the form of a harmonic oscillator counting the number of bosons.

If we'd want to consider a real quantum dot, we'd have to remain general and treat all low order V_jk as parameters - we cannot simply put the V_jk with j<2 or k<2 to zero as you propose, since these terms describe how the empty dot and the dot filled with one particle respond to the e/m field.

But the purpose of the exercise is to learn about standard Lagrangian field theory in a very simplified setting. Therefore we keep things as simple as possible subject to the constraint that we have an associated Lagrangian. Thus, in order to be able to use the quantum dot as an example for illustrating the dynamics of Wightman fields, we choose the interaction artificially so that it corresponds to a system derived from a standard quartic (or if this seems too complicated for you, cubic) Lagrangian.

Then these additional terms cannot have any effect on the 0-boson (=empty vacancy) and 1-boson (vacancy+one boson) states. So, these terms *cannot* have the structure ~q^3 + q^4. I propose interaction ~a*a*aa as a reasonable alternative. If the non-quadratic terms are some corrections to the boson-crystal interactions, then why didn't we took them into account in our original formulation of the Hamiltonian (1)?

Instead of simply solving the exercise and learning from it, you turn it into a long discussion about the modeling of a real quantum dot. This is like refusing to solve an exercise in classical mechanics on the anharmonic oscillator because a real oscillator has an extension, friction, and all that stuff, while the purpose of the exercise is simply to get practice in a certain way of thinking.

I want to teach you intuition about Wightman fields, not about quantum dots!

 

[Physics Monkey]

I don't find this reference to solid state physics convincing. In solid state physics you have a "medium", e.g., a crystal. In the quasiparticle picture you don't treat this medium explicitly, but model its presence implicitly by renormalizations. If I am interested in an electron moving through empty space, there is no "medium" of any kind. Likewise, there is no "medium", when I am studying particles placed in a small box. These two cases should not involve renormalizations.

Eugene.

Why are you so sure that "empty space" isn't a kind of medium? Do you think that if you lived in a medium as in solid state physics that you would know it? Wouldn't you be just as sure that "empty space", by which you would mean the state containing no low energy excitations, was a boring place even if it wasn't?

A nice example of this kind of medium view occurs in QCD. There one can imagine producing a high energy quark, for example, that behaves more or less as a free particle because of asymptotic freedom. This quark then loses energy, and as it does, it begins to interact more strongly. And as it interacts more strongly, it deforms the quantum state into something that eventually looks like a bunch of hadrons flying off at high speed. This process is a lot like the turning on of interactions that can be accomplished in a controlled way in condensed matter physics.

 

[meopemuk]

I want to teach you intuition about Wightman fields, not about quantum dots!

I find your "quantum dot" or "particles-in-a-box" model very conterintuitive, because it implies some non-zero interaction energy even in the case when there are no particles at all, i.e., nothing can interact. Perhaps we can agree that your pedagogical device did not achieve its purpose of intuition-building and move on.

Eugene.

 

[meopemuk]

Why are you so sure that "empty space" isn't a kind of medium?

Of course, I cannot be sure. I simply choose the "empty space" model for aesthetical reasons, because it is simpler than the background "medium" model. In experiments we see degrees of freedom associated with particles. So, it is reasonable to assume that there are no other degrees of freedom, i.e., those associated with "media" or "fields".

Moreover, it appears that QFT effects usually attributed to the influence of the "medium" (such as "vacuum polarization" or "electron self-energy" radiative corrections) can be explained simply by modification of potentials acting between particles moving in empty space. See https://www.physicsforums.com/showthread.php?t=474666 [/URL].

Eugene.

 

[A. Neumaier]

I find your "quantum dot" or "particles-in-a-box" model very conterintuitive, because it implies some non-zero interaction energy even in the case when there are no particles at all, i.e., nothing can interact.

The dot is a physical object which changes its state when an interaction is switched off. This is the case both when the dot is empty and when it is occupied. What is conterintuitive about that?

But even if you find the model very artificial, it is needed to demonstrate in a simple situation the meaning of the Hamiltonian in the Wightman representation. So if you want to understand the latter, I invite you again to do the exercise, in the form specified earlier.

 

[meopemuk]

The dot is a physical object which changes its state when an interaction is switched off. This is the case both when the dot is empty and when it is occupied. What is conterintuitive about that?

What I fail to understand is the origin of interaction described by the non-quadratic terms in your Hamiltonian. What interacts with what there?

(a) this is interaction between particles in the box (=dot), which was ignored in our original quadratic Hamiltonian

(b) this is interaction between a single particle and the box (=dot), which we failed to include in the original quadratic Hamiltonian

(c) this is interaction with external electric and/or magnetic field.

Eugene.

 

[A. Neumaier]

What I fail to understand is the origin of interaction described by the non-quadratic terms in your Hamiltonian. What interacts with what there?

(a) this is interaction between particles in the box (=dot), which was ignored in our original quadratic Hamiltonian

(b) this is interaction between a single particle and the box (=dot), which we failed to include in the original quadratic Hamiltonian

(c) this is interaction with external electric and/or magnetic field.

(c) is right. The quadratic Hamiltonian (case g=0) describes an ideal quantum dot in which the added particles are assumed not to interact. The nonquadratic terms (linear in g; quartic in q) describe the effect of the external field. They affect the dot, the particles in the dot, and the way they interact. All this should have been apparent from the way I derived the Hamiltonian.

Removing the shift of energy of the empty dot due to the external field is the vacuum renorrmalization = zero point energy subtraction by normal ordering. Adapting the particle mass to match the difference between the energies of empty dot and singly occupied dot when the external field is applied gives the mass renormalization. Both are physical = measurable effects.

 

[meopemuk]

(c) is right. The quadratic Hamiltonian (case g=0) describes an ideal quantum dot in which the added particles are assumed not to interact. The nonquadratic terms (linear in g; quartic in q) describe the effect of the external field. They affect the dot, the particles in the dot, and the way they interact. All this should have been apparent from the way I derived the Hamiltonian.

Removing the shift of energy of the empty dot due to the external field is the vacuum renorrmalization = zero point energy subtraction by normal ordering. Adapting the particle mass to match the difference between the energies of empty dot and singly occupied dot when the external field is applied gives the mass renormalization. Both are physical = measurable effects.

OK, so now we have three physical systems that interact with each other non-trivially. There is the crystal medium, which is represented by the confining potential acting on particles and changing under the influence of the external field. There is a system of n particles, which interact with other, with the confining potential and with the external field. And there is the external field. When the field is off the confining potential is quadratic and the energy levels (n-particle states) are equidistant. When the field is on, the confining potential acquires non-quadratic corrections, and the energy levels are not equidistant anymore. I hope I get it right.

Now, all this theory is written in terms of a/c operators a* and a of particles defined for the field-off situation. When we turn the field on, these operators no longer describe individual particles. The new "physical" particles are complex linear combinations of the previous "bare" particle states. The new 0-particle vacuum is also a complex linear combination. Despite what you said, this new situation cannot be remedied simply by the energy shifts of the new 0-particle and 1-particle states. We need to find the "physical" vacuum and the "physical" 1-particle state and new "physical" a/c operators as functions of the old "bare" stuff.

Are you saying that I need to go through all these hoops just to get intuition about the time evolution of one or two pesky particles? And this is even before you allowed the particles to move in space? My poor intuition is going to explode if I not bail out right this moment.

Regards.
Eugene.

 

[A. Neumaier]

OK, so now we have three physical systems that interact with each other non-trivially. There is the crystal medium, which is represented by the confining potential acting on particles and changing under the influence of the external field. There is a system of n particles, which interact with other, with the confining potential and with the external field. And there is the external field. When the field is off the confining potential is quadratic and the energy levels (n-particle states) are equidistant. When the field is on, the confining potential acquires non-quadratic corrections, and the energy levels are not equidistant anymore. I hope I get it right.

Yes. That's the description of an idealized quantum dot occupied by bosons.

Now, all this theory is written in terms of a/c operators a* and a of particles defined for the field-off situation. When we turn the field on, these operators no longer describe individual particles.

Of course. a and a^* create free particles, not interacting ones, since they were constructed from the free Hamiltonian, not from the interacting one.

The new "physical" particles are complex linear combinations of the previous "bare" particle states. The new 0-particle vacuum is also a complex linear combination. Despite what you said, this new situation cannot be remedied simply by the energy shifts of the new 0-particle and 1-particle states. We need to find the "physical" vacuum and the "physical" 1-particle state and new "physical" a/c operators as functions of the old "bare" stuff.

If you like, you can work out (along the lines of your book) the dressing transform that transforms the free creation operator a^* into the creation operator a_g^* creating a dressed 1-particle state from the dressed vacuum when the coupling constant is g. Expressed in the dressed operators, your dressed Hamiltonian will have the form you want. But this complicates the calculation, since you must then also work out the representation of the original q(t) on the Fock space determined by the dressed operators acting on the dressed vacuum. All in all, it more than doubles the work.

If you need it to understand what is going on, well, then please do it both ways - so that you can see how things look both from the Wightman field perspective and from the dressed particle perspective. Perturbatively, everything is well-defined in both approaches since you'll not encounter any divergences. (This is the advantage of considering a quantum point rather than particles moving in space.) to Moreover, the observable consequences will be the same since the dressing transformation is just a change of basis, not a change of the physics.

Are you saying that I need to go through all these hoops just to get intuition about the time evolution of one or two pesky particles? And this is even before you allowed the particles to move in space? My poor intuition is going to explode if I not bail out right this moment.

Nothing will explode, just be patient. The exercises are not difficult - you could even do them purely mechanically -, and if you do it in the form I suggest (rather than transforming the interacting problem to the dressed version) the calculations are fairly short. But you need to go through all this in order to understand the real meaning of Wightman functions and the corresponding time evolution. Building the intuition that you are currently lacking completely takes some practice.

Once the quantum point is fully understood, you'll have enough intuition so that we can do moving particles in a more roundabout fashion.

 

[meopemuk]

Of course. a and a^* create free particles, not interacting ones, since they were constructed from the free Hamiltonian, not from the interacting one.

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.

If you like, you can work out (along the lines of your book) the dressing transform that transforms the free creation operator a^* into the creation operator a_g^* creating a dressed 1-particle state from the dressed vacuum when the coupling constant is g. Expressed in the dressed operators, your dressed Hamiltonian will have the form you want. But this complicates the calculation, since you must then also work out the representation of the original q(t) on the Fock space determined by the dressed operators acting on the dressed vacuum. All in all, it more than doubles the work.

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. The desirable way is to define the Hamiltonian, so that there are no unphysical self-interactions in the vacuum and 1-particle states from the outset. 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.

Unfortunately, Hamiltonians in existing QFT theories do not obey this principle. I firmly believe that disregarding this important principle of no-self-interaction is the source of most problems and confusions characteristic for traditional QFT. So, in order to correct the bad QFT Hamiltonians one needs to apply this messy "dressing" procedure to fix all the renormalization and divergence problems. So, I use the "dressing" procedure reluctantly in order to make a connection to well-established but ill-formulated previous theories, like QED. Ideally, the Hamiltonian would be formulated without self-interaction terms and the "dressing" would not be needed.

Now you suggest to consider a theory, which is formulated in this inappropriate non-transparent self-interacting way from the beginning. 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. I believe that this is the true Hamiltonian, which can be used in routine quantum mechanical calculations without any tricks and renormalizations. 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".

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.

Eugene.

 

[meopemuk]

(c) is right. The quadratic Hamiltonian (case g=0) describes an ideal quantum dot in which the added particles are assumed not to interact. The nonquadratic terms (linear in g; quartic in q) describe the effect of the external field. They affect the dot, the particles in the dot, and the way they interact. All this should have been apparent from the way I derived the Hamiltonian.

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. If this is so, then each particle feels exactly the same external potential/field. If the particles don't interact between themselves, then the total energy is going to be proportional to the number of particles E=ne. So, the only way to get a non-equidistant spectrum is to allow the interaction between particles in the dot. Then we are back to the same old question: how can it be that interaction between particles affects the energy of the no-particle state?

Another possibility to have a non-equidistant spectrum is to assume that the confining potential and/or the external field depends on the number of particles placed in the dot. This would be a strange model, indeed.

Eugene.

 

[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.