# Why isn't the Segal-Bargmann space used more often in Quantum Mechanics?

The Stone-von Neumann Theorem states that a Hilbert space on which is defined an irreducible set of operators which satisfy the exponentiated canonical commutation relations is unitarily equivalent to $$L^2(R^n)$$.

One such space is the Segal-Bargmann Space, which is the space of holomorphic functions that have finite norm under the inner product given by:

$$(F,G) = \int_{C^n} \overline{F}(z) G(z) e^{-|z|^2} \mathrm{d}z$$

As explained in the Wikipedia article elements of this space can be thought of as functions on phase space (since $$C^n$$ is $$2n$$-dimensional) which seems like a more (or at least as useful) description of wave functions as in $$L^2(R^n)$$, given that Quantum Mechanics is based on Hamiltonian formalism.

Also the fact that functions are holomorphic seems like it's easier to define differential operators (rather than using a dense subspace). I've just finished taking an undergraduate course in Quantum Mechanics but it seems to me like this space a lot of advantages but I've never heard it mentioned in a lecture or in any common QM book. Why is this so? Is there something I'm missing which actually makes this space more complicated to use?

• It seems that you speak about holomorphic representation, doesn't it? As I remember, Zinn Justin considers holomorphic representation for oscillator with perturbation in his book "path integral in QM". He provides holomorphic description of QHO and uses it to construct path integral. Jan 11, 2020 at 19:42
• Also this question may be useful, physics.stackexchange.com/q/362847 Jan 11, 2020 at 19:45
• It could simply be that, once all of the literature is written using a certain mathematical formalism, it takes too much effort to cast it in another formalism, and the fact that there aren't many people experienced with the formalism means that peer review of such a textbook would be more difficult. Jan 11, 2020 at 19:48
• In addition, there is a book "oscillator representation in quantum physics" by Efimov, where he describes foundations of these representation Jan 11, 2020 at 19:49
• As Artem comments, physicists often call this the "holomorphic representation" and it is used in physics, just not in undergraduate textbooks. I'm not sure this question has an answer rooted in physics rather than history and opinion. Jan 11, 2020 at 19:51

The reason is that, in my view, this representation is not the representation of any observable: there is no observable which is multiplicative in this representation. Instead, the creation and annihilation operators (which are not observables) take a simple form. Therefore this representation is useful in contexts where those operators play an important role, e.g., in qft. The path integral theory can fruitfully developed in this formalism. There is a book that adopts this formulation (authored by Faddeev and Slavnov if I correctly remember).

• The lopsided (non-diagonal) Wigner functions in phase space quantization for the oscillator, as you point out, are elegantly simple in this language, a fact used copiously by mathematicians (Grossmann, Loupias & Stein, 1968) who breathe complex variables. Jan 11, 2020 at 20:49

They are used extensively in so-called coherent state (and vector coherent state) representations, albeit the form of the inner product that you give is bypassed for explicit calculations. Indeed the primary interest of this is the simplicity of representing creation and destruction operators in terms of differential operators and multiplication by Bargmann variables.

The technique was reviewed in

Rowe, D. J. "Dynamical symmetries of nuclear collective models." Progress in Particle and Nuclear Physics 37 (1996): 265-348.

and the antecedent

Rowe, D. J. "Microscopic theory of the nuclear collective model." Reports on Progress in Physics 48.10 (1985): 1419.

This is fairly hardcore stuff as the representation theory of the non-compact symplectic group $$\mathfrak{sp}(6,\mathbb{R})$$ is not easy. As a result this is not so popular although the reviews above are fairly well cited (over 100 and 200 times respectively, according to GoogleScholar). Similar techniques by the same group have been applied to obtain irreps of $$\mathfrak{so}(5)$$ and others.