I did all the courses on Quantum Mechanics and QFT which my faculty offers and up to now no one defined to me what a Heisenberg-Weyl algebra actually is.

This appears in my studies when studying the polymer representations in QM.

So the points not clear to me are:

  1. What is and why is it important the Heisenberg-Weyl Algebra in QM? I read that is a kind of operator algebra but it was very murky and unclear to me.

  2. What is the role of the usual $\hat{X}$ and $\hat{P}$ operators here?

  3. Regard to that Algebra, do the $\hat{a}$ and $\hat{a}^\dagger$ have a special role or, as it was told me, they are just "nice" operator since they create quanta?

A good math oriented answer, defining things precisely, will be more appreciated.

  • 3
    $\begingroup$ This might be a good handle... $\endgroup$ Apr 11 at 22:29

1 Answer 1


Given a real symplectic space $(X,\sigma)$ with non-degenearated symplectic form $\sigma: X\times X \to \mathbb{R}$, there is a unique (up to $C^*$-isomorphisms) unital $C^*$-algebra $W(X,\sigma)$ generated by the distinct generators $W(f)$ for all $f\in X$ such that $$W(f)^*= W(-f)\quad \mbox{and}\quad W(f)W(g) = e^{i\sigma(f,g)/2} W(f+g)\:.$$ That is the Weyl-Heisenberg ($C^*$) algebra of $(X,\sigma)$.

As a consequence, the elements of $W(X,\sigma)$ are (generally finite) linear combinations of products of elements $W(f)$ (i.e. linear combinations of the elements themselves as a consequence of the second identity above). A $C^*$-algebra is equipped with a unique norm satisfying the C* condition $||a^*a||=||a||^2$ and the above infinite linear combinations are the ones that converge with respect to that norm. This is the meaning of "generated" here.

A concrete case. Take $X= {\mathbb R}^2$ with coordinates $(x,y)$ and the symplectic form (*) is $$\sigma((x,y),(x',y')) = xy'-yx'\:.$$ In this case $W((X,\sigma))$ is isomorphic to the $C^*$-algebra of bounded everywhere defined operators generated by the unitary operators $$\hat{W}((x,y)) := e^{i(y\hat{X}-x\hat{P})}\quad (x,y)\in \mathbb{R}^2$$ where $\hat{X}$ and $\hat{P}$ are the standard position and momentum operators in the Hilbert space $L^2(\mathbb{R},dx)$.

Decomposing $\hat{X}$ and $\hat{P}$ in terms of $a$ and $a^\dagger$, you have an equivalent definition of the operators above.

Technical comment: $y\hat{X}-x\hat{P}$ in the exponent is actually the closure of the linear combination $y\hat{X}|_{S}-x\hat{P}|_S$, where $S= {\cal S}(\mathbb{R})$ is the Schwartz space. These linar combiantions are essentially selfadjoint so that the closure is a selfadoint operator as it is due.

The position and momentum operators are here obtained back as the (unique) generators of the families (one-parameter groups of unitary operators) of elements $\hat{W}((0,y))$ and $\hat{W}((x,0))$.

As a reference see, e.g., one of my books.

This algebra is important because it is one of rigorous ways to speak of canonical commutation rules. Also in QFT, where the symplectic space is the one of solutions of the classical filed equations equipped with the standard invariant symplectic form and the exponent are the abstract quantum fields smeared with solutions as in the LSZ formalism. The strategy to deal with bounded everywhere-defined operators avoids all problems with domains of unbouded operators.

(*) Check signs and coefficients $1/2$ please.


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