Exploring the Strange Link Between Quantum Mechanics & Quantum Field Theory

[Eye_in_the_Sky]

In that case, if you are interested, I will try to post something (just presenting the main points at first; I will get more detailed if anybody is interested).

As I said, I (for one) am interested. And I agree, a "generalized sketch" is an appropriate way to start. Afterwards it can be evaluated whether or not and just how to continue.

... And I see you've already begun to sketch things out:

One starts from the desire of having a many-body description and the introduction of the Fock space and of annihilation/creation operators. From the basic meaning of those operators, their commutation relation is deduced. The field is introduced as a *byproduct*, not as a starting point. And there is no classical field introduced, one gets directly to the quantum fields. The CRs between the field and conjugate momenta *follow* from the Crs (or antiCRs) between the creation/annihilation operators and these are introduced very naturally (imho), being due to the very nature of the creation/annihilation operators.

... The nonrelativistic case is handled exactly the same way as the relativistic case. The only difference is the need to impose that one-particle state created by the field operator have definite transformation properties under Lorentz transformations. For example, for a real KG field, this forces one to combine annihilation and creation operators (of positive energy states) together, which was not required in the non-relativistic case (and this leads to the fact that in the KG case, the momentum conjugate to Phi is the time derivative of Phi whereas it is Phi^* in the nonrelativistic case).

I already have some questions. But for now I am going to hold back on them.
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I am not sure what you mean by "advanced tools".

I mean the 'tools' which need to be introduced in connection with the non-free field.

By what you have said, it seems to me that a knowledge of those 'tools' will not be required.
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My question was about the "deep" connection between the CRs of the field and conjugate momentum and the CRs of the annihilation/creation operators.

So, I think then we can all see and agree that – as far the nonrelativistic case is concerned – with regard to this connection, there is nothing "deep".

If we perform a quantization of the field (see posts #25 and #26), then the CRs for Ψ and Π (in light of Π = iΨ^†) simply tell us that Ψ^† and Ψ are already themselves no more (and no less) than creation and annihilation operators. (As for their dependence on time, this merely reflects the fact that they are now in the "Heisenberg picture".)

In fact, if we go back to an arbitrary wavefunction ψ(x) which solves the time-dependent Schrödinger equation, and write it as

ψ(x) = ∑_n a_n(t) φ_n(x)

with respect to some orthonormal basis {φ_n(x)}, then a little bit of reflection shows us that "quantization of the field" is the same as invoking a direct canonical quantization upon the very coefficients a_n(t) themselves and their complex conjugates, which when promoted to operators become the 'familiar' annihilation and creation operators (now expressed in the "Heisenberg picture").

In other words, upon a little more reflection, we see that:

Annihilation and creation operators are precisely what one gets when one "quantizes" probability amplitudes and their complex conjugates.

And furthermore, from this, we can then see that the reason why, in the relativistic case, the fields Ψ^† and Ψ cease to have the character of creation and annihilation operators is, quite simply, because at the 'classical' level the field ψ itself ceases to have the character of a probability amplitude.

Next ... for the nonrelativistic case, what are the "mode" functions with respect to which creation and annihilation operators are defined? ... Well, a complete set of "modes" can be understood as nothing but the complete set of eigenfunctions of some observable on the single-particle Hilbert space. For example, if we choose X, then we get the field operators Ψ^†(x) and Ψ(x) (now, in the "Schrödinger picture"). If we choose P, we get field operators
Φ^†(p) and Φ(p). If we choose the single-particle Hamiltonian H, then we get operators a_E^† and a_E. And on it goes ...

... So, I wonder now how all of this works in the relativistic case?

Patrick, can you tell me?

My thoughts on this have evolved a bit. In Weinberg's approach, the basic CRs are simply between the annihilation/creation operators. The CRs between the fields and conjugate momenta are simply wasy to encompass this information.

Thinking in terms of independent degrees of freedom, it seems to me now that the conventional approach is simply a way to isolate the independent degrees of freedom. It is instructive to see that for an equation linear in time, the conjugate field momentum is the complex conjugate of the field (for the nonrelativistic Schrodinger case or for the Dirac equation) whereas it is the time derivative for the real KG field, and it's the complex conjugate of the time derivative of the field for the complex KG field. But in the end, th CR's between the field and the conjugate momentum always boils down to isolating the creation and annihilation operators and specifying the CRs between them (by CRs here I include antiCRs, i.e. I am talking in the most general context). So I think that the field approach is just a clever way of isolating the independent degrees of freedom and giving their CRs which, in the end, always amounts to specifiying the CRs between the annihilation and creation operators.

So, looking only at the CRs, it seems to me that the field approach is a very indirect (but very clever) way of giving the CRs between the annihilation and creation operators. The conjugate momentum keeps changing definition but in the end, the CRs between the field and conjugate momentum always ends up giving the same CRs between the annihilation/creation operators.

The real advantage of the field approach is that it automatically builds the fields with the correct Lorentz properties, whereas in the Weinberg approach, some work is required to find the right combination of annihilation and creation operators that will do that.

I look forward to hearing more from you about this, starting with a "generalized sketch" ... that is, if you aren't having 'second thoughts'.
[I know that "time" is a valuable and, often (ha!ha!), scarce resource.] Please, let me know if you are (or if you are not) going to make an attempt.

In the mean time, I want to go back and comb through all of the posts so far in this thread in order to pick out some more of the many points which have been made that can use further clarification.
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I still have a question about your comment concerning the action for Schroedinger's equation but I will post later.

So, what is it?

 

[Eye_in_the_Sky]

recovering X & P from the quantized Schrödinger-field

After combing through the various posts of this thread, I have found several points which, it seems to me, could use some further clarification.

Here is one of them:

Starting from the quantized Schrödinger-field, how does one recover the single-particle description in terms of X and P? What connection, if any, exists between the CRs (or anti-CRs) of the quantized fields and the CRs of the single-particle operators?

Back in the last part of post #26, it was shown that we can make the identification

Ψ^†(x,0)|0> = |x> ,

the eigenkets of X, the single-particle position operator in the "Schrödinger picture".

A similar derivation, with the time-dependence retained in Ψ^†, shows that we can make the identification

Ψ^†(x,t)|0> = |x,t> ,

the eigenkets of X(t), the single-particle position operator expressed in the "Heisenberg picture". Therefore, we can write

X(t) = ∫ x Ψ^†(x,t)|0><0|Ψ(x,t) dx ... [1] .

This identification follows directly from the CRs (or anti-CRs) of the fields in connection with their equations of motion.

It follows in two steps:

1/ The CRs (or anti-CRs) of the fields are what allow us to compute for the dynamics of the fields; that is, they allow us to convert the CRs on the right-hand-sides of the relations (see post #25)

i∂_tΨ = [Ψ,H]
i∂_tΠ = [Π,H] ,

where H = -i ∫dx ΠH_xΨ ,

into

[Ψ,H] = H_xΨ
[Π,H] = -H_xΠ .

From here, we can write the equations of motion for the fields:

i∂_tΨ = H_xΨ
i∂_tΠ = -H_xΠ .

2/ Once those equations of motion have been given and solved for, the CRs (or anti-CRs) of the fields will then serve again. They will now tell us that Ψ^† (remember that Π = iΨ^†) and Ψ have the character of creation and annihilation operators. And from this, the identification of X(t) according to [1] will follow.

Next, once we have identified X(t) as in [1], what about P(t)?

By definition P(t) is the momentum conjugate to X(t). That is, by definition
P(t) is an operator on the single-particle Hilbert space such that

[X(t),P(t)] = i , for all t .

This relation has nothing to do with the CRs (or anti-CRs) of the fields.

But ... so what? At this point we don't even need P(t)! The only reason we ever had for the introduction of P(t) was in order to solve for the single-particle dynamics. But this we have already accomplished by having solved for the dynamics of the fields.

On the other hand, if we do decide to introduce P(t), what then will our field have to say?

... Well, from the form of the equation of motion of the field,

i∂_tΨ(x,t) = [-∂_x²/(2m) + V(x)]Ψ(x,t) ,

it will follow that an acceptable choice for P(t) is mdX(t)/dt.