# Plausibility of the Weyl-Relations of position and Momentum - Physical meaning of the Heisenberg group

In this question I asked about the uniqueness of the momentum operator $$\hat{p}$$ for a given position operator $$\hat{x}$$, and wether the uniqueness was fixed by the commutation relations that position and momentum have to satisfy. The answer pointed out that uniqueness was not given alone by the commutation relations, but instead if I require them to hold in an exponential form (the so called Weyl-Relations): \begin{align} e^{i\hat{x}t}e^{i\hat{p}s}=e^{-its}e^{i\hat{p}s}e^{i\hat{x}t} \end{align} With $$s$$, $$t$$ being real numbers. It was further stated that if the operators satisfy the Weyl-Relations, then they would as well generate the Heisenberg group.

What I know now is that either the demand to represent the Heisenberg group or the demand for the operators $$\hat{x}$$ and $$\hat{p}$$ to satisfy the Weyl Operations uniquely defines one of the operator, if the other one is given. This leads me to my question: Is there any plausibility of the Weyl Relations, or any interpretation that is connected with this relations? I'm looking for an intuitive reason for the operators to satisfy this relations, besides the obvious reason "It works".

I could ask the same about the Heisenberg group: Why should it be intuitive that a set of observables is somehow connected to this group? Has this to do with symmetry arguments? For example, could it be that certain symmetries only can be represented if $$\hat{x}$$ and $$\hat{p}$$ don't generate the Heisenberg group?

To give an example what would be a satisfactory answer to me: In electrodynamics we have property of the equations being linear. While this is a pretty mathematical statement, it translates to the very intuitive and plausible concept that one can superimpose two solutions, and the result is a solution again. It's be nice of something similar could be pointed out about the Weyl Relation / the Heisenberg Group.

• – Cosmas Zachos Nov 23 '18 at 1:35

## 1 Answer

I am not sure what rates as satisfactory intuition or not, for QM.... Weyl in his 1927 paper appreciated that Heisenberg's Lie algebra should exponentiate to a group, that of translations and locations of a clock with n hours, the toy archetype of Hilbert space and took the limit of infinite hours with a suitable scaling—the "wrong" scaling would net $$SU(\infty)$$, thus Poisson Brackets and classical mechanics, not the QM Hilbert space whose motions are studied here.

I can't avoid the "just so" pitfall, but let me try to arrange the facts, hoping it might let their reflections off each other sharpen.

As the ringleader of the 20s Gruppenpest charge, Weyl investigated the group resulting from the exponentiation of the Lie algebra, $$[\hat x, \hat p]=i\hbar ,$$ so with generic unitary group elements $$e^{it\hat x + is\hat p+i\hbar u} = e^{i(u+st\hbar/2) } e^{it\hat x} e^ {is\hat p } ,$$ with the sufficient eponymous braiding relations (group element multiplication identities, (26) in Weyl op.cit.) you are interested in, $$e^{it\hat x} e^ {is\hat p } = e^{-its\hbar} e^ {is\hat p } e^{it\hat x} .$$ (Sufficient in the sense that differentiating them w.r.t. s and t and setting these parameters and u equal to 0, excursion off the origin, retrieves the Lie algebra commutation relation.) These identities, straightforward consequences of the CBH composition rules of the Heisenberg algebra, comprise all you need to know to evaluate group actions on QM vectors. The two chosen group elements suffice to specify the entirety of phase-space behavior of quantum mechanics.

They act unitarily on functions, e.g., of x, as $$e^ {is\hat p } \psi(x)= \psi(x+\hbar s) ,$$ the Lagrange shift operator, (51) in Weyl. For instance, the conserved Noether charge in a translationally symmetric theory would yield that momentum symmetry generator.

A less familiar action is that of clock-reading rephasing, (50) in Weyl, $$e^ {it\hat x } \psi(x)= e^{itx} \psi(x),$$ and, of course, the constant rephasing by iu through the central element.

At this level, I really don't have much intuition for the latter, beyond Weyl's original "behold!" analogy in his section 8, to the finite clock matrix $$\Sigma_3$$ of Sylvester (1882), a diagonal matrix with a constant phase in each entry augmenting its phase ordinally. You might think of a clock with exponentiated hour-phases. The translation operator is, of course, $$\Sigma_1$$, the shift matrix by an hour. In a careful infinite-hour limit, one recovers his operators, and the logarithms of these operators, (yes, there is an ingenious method of taking them), retrieves the Heisenberg algebra, by a celebrated argument of Santhanam & Tekumalla 1976 ; pure magic: the traceless r.h.side becomes the non-trace-class identity! But that's a whole different fascinating question.

I think Weyl always cherished this intuition (he rhapsodized it in his abovelinked book), and it has really helped me in several logical quandaries... You might, or might not, give it a try...