Are there any theorems that support the commutation relations in QFT?

Question

I am learning Quantum Field Theory. I am confused about the commutation relations, which says that the field and its conjugate momentum don't commute. Are there any theorems that support the commutation relations in QFT? Or why the relation is correct? Simply due to experimental test?

Answer 1

The main opinion of the physical community today is that it's an axiom / rule of nature, e.g. cannot be explained any further. However, there are other standpoints that can give an equivalent description but a different interpretation of quantum mechanics.

What the commutation relations do, is essentially the introduction of a principle of uncertainty, namely when $[q, p] = i \hbar$ then $q$ and $p$ cannot be measured simultaneously. In experiments we observe that a measurement of $q$ will alter $p$ and vice versa. The commutation relation is an expression of that property.

The question is therefore: Is this principle of uncertainty explainable in terms of different assumptions or physical observations which are more general?

For particle mechanics this is not the case, since particles, in the classical sense, have always a well defined position and momentum. The principle of uncertainty has to be put in by hand to account for the experimental data. Every other and equivalent formulation of quantum particle mechanics has some hidden or obvious way of imposing the principle.

Yet, we already know that quantum particle mechanics is not the ultimate theory. A more general and successful theory is quantum field theory (QFT) which describes nature in terms of fields. Our current formulation also imposes the principle of uncertainty by hand in a similar way as we did it in quantum particle mechanics (namely by the commutator relations). However, every equation of motion of quantum field theory (that we currently think is physical) is a wave equation (with small correction terms). Waves have an inherent uncertainty property that forbids simultaneous measurements of field amplitudes and generalized momentum amplitudes, e.g. you cannot measure a wave amplitude (with another wave) without altering the wave's momentum. So there is a chance that this principle of uncertainty can be explained in a more general way.

My view on this is described here

Answer 2

Indeed we have to implement the CCR or CAR as an axiom to QM/QFT. But if you like to think about it in terms of an action principle look at Schwinger's quantum action principle. From this axiom you can derive the relations. See here

Basically you start with the operator valued action $S$:

$$S = \int_{t_1}^{t_2} dt \frac{1}{2}\sum_i p_i\dot{q}_i +\dot{q}_i p_i - H(q,p,t)$$

and use infinitesimal unitary transformations of $q,p,t$ to derive $\delta S$. The main point is now to say, that $\delta S = G(t_2)-G(t_1)$. $$ \delta S_{21} = \int_{\tau_1}^{\tau_2} d\tau \big{[} \frac{1}{2} \sum_{i=1}^n \delta\tilde{p}_i \frac{d \tilde{q}_i }{d\tau} + \frac{d \tilde{q}_i }{d\tau}\delta\tilde{p}_i -\delta\tilde{q}_i \frac{d \tilde{p}_i }{d\tau} - \frac{d \tilde{p}_i }{d\tau}\delta\tilde{q}_i -\delta\tilde{H}\frac{dt}{d\tau}+\tilde{H}\delta\frac{dt}{d\tau}\big{]}+ \int_{\tau_1}^{\tau_2} d\tau\frac{d}{d\tau}(\frac{1}{2}\sum_{i=1}^n(\tilde{p}_i\delta \tilde{q}_i + \delta\tilde{q}_i \tilde{p}_i)-\tilde{H}\delta t)$$

From this equation we get an expression for the generator $G(t)$ of this unitary transformation. $$G = \frac{1}{2}\sum_{i=1}^n(\tilde{p}_i\delta \tilde{q}_i + \delta\tilde{q}_i \tilde{p}_i)-\tilde{H}\delta t $$ Now it is easy to compute the CCR and CAR by using the $\delta B = \frac{i}{\hbar}[G,B]$ for any operator $B$ (for example $x_j,p_j$).

Answer 3

It is just a way to get from a classical to what we call a quantum theory. In classical mechanics you have the poisson brackets governing the dynamical behaviour of your system. If you leave the Hamiltonian/Lagrangian the same and change this poisson bracket to a commutator of operators you obtain a new theory. This is than basically labeled "Quantum". There is of course a whole array of material on why this way of quantizing a theory is identical to other ways like for example the path integral formulation.

Of course in the end the only way to know that this gives you something "correct" is experimental. In fact you can create a lot of theories which are consistent but ultimately probably don't give a description of nature. It just so turns out that experiments confirm up to some point that the theories which do describe nature in particular QED are theories which are in the sense Quantum that its components, the fields and their conjugate moment, fullfill canonical commutation or anticommutation relations.

As a side node the mathematics behind QFT is partially not rigorous mathematics at all. Completely rigorous mathematical QFT has already a problem with operator valued distributions, which fields supposedly are, because they commute to the delta function. In this sense you have trouble even defining a field, but just because the mathematical tools aren't there doesn't necessarily mean the theories are invalid.