User:Lethe/Heisenberg

The Heisenberg group and the associated Schwartz space

[1]

Realization by Fock

Define an inner product on the space of entire functions on C

${\displaystyle (f,g)=\sum _{m}m!{\overline {\alpha }}_{m}\beta _{m}}$

where

${\displaystyle f(z)=\sum _{m}\alpha _{m}z^{m}}$

and

${\displaystyle g(z)=\sum _{m}\beta _{m}z^{m}.}$

Define operators

${\displaystyle [A^{*}f](z)=zf(z)}$

and

${\displaystyle [Af](z)=f'(z).}$

Then ${\displaystyle A^{*}}$ and ${\displaystyle A}$ satisfy the canonical commutation relations

${\displaystyle [A^{*},A]=1}$

and are at least formal adjoints. This description is due to Fock.[2] In the 1960s, Bargmann [3] realized this inner product as a Lebesgue space with a Gaussian measure:

${\displaystyle (f,g)=\pi ^{-n}\int f(z)g(z)e^{-|z|^{2}}\,dz,}$

for a system with n degrees of freedom. Segal simultaneously and independently defined the holomorphic realization for systems with infinitely many degrees of freedom [4] and Bargmann subsequently also treated the infinite degrees of freedom case[5]. One thus refers to the BFS construction for Bargmann, Fock, and Segal.

Heisenberg group

Let (V,ω) be a symplectic vector space. The Heisenberg group H(V) associated to V is a nilpotent real Lie group whose underlying manifold is V×R and a group law given by

${\displaystyle (v_{1},t_{1})\cdot (v_{2},t_{2})=(v_{1}+v_{2},t_{1}+t_{2}+{\frac {1}{2}}\omega (v_{1},v_{2})).}$

The Lie algebra ${\displaystyle {\mathfrak {H}}(V)}$ can be canonically identified with the manifold, so to distinguish between elements of the Lie algebra and the group, use exponential notation.

An admissible complex linear structure on V is a linear complex structure J such that

${\displaystyle \omega (Jv_{1},Jv_{2})=\omega (v_{1},v_{2})}$

and

${\displaystyle \omega (v,Jv)\geq 0.}$

Denote by VJ the complex vector space induced by J. Then we have a Hermitian form.

According to the Stone-von Neumann theorem, for each λ > 0, there is a unique (up to equivalence) irreducible representation ${\displaystyle (\pi ,{\mathcal {H}}_{\pi })}$ of H(V) satisfying

${\displaystyle \pi (e^{tE})=e^{i\lambda t}}$

Bases

Let {en}n be an orthonormal basis for VJ. Then define {fn}n by fm = Jem. {e,f} is then a basis for V, called a standard basis and e and f are called canonically conjugate vectors. Note that ω(ei,fj) = δij.

Defining

${\displaystyle a_{m}^{\pm }={\frac {1}{\sqrt {2}}}(\mathbf {e} _{m}\mp i\mathbf {f} _{m}),}$

we have that a± give a basis for the complexification V± and {a+,a} give a basis for VC, the standard basis (induced by {e,f }). Viewing a+ and a as elements of VC with the symplectic form extended by linearity, we have

${\displaystyle \omega (a_{i}^{+},a_{j}^{-})=i\delta _{ij}}$

whereas if they are viewed as elements of ${\displaystyle {\mathfrak {H}}(V)}$ then we have the commutation relations

${\displaystyle [a_{i}^{+},a_{j}^{-}]=i\delta _{ij}E.}$

The Schwartz space

To each representation ${\displaystyle (\pi ,{\mathcal {H}}_{\pi })}$ we can associate a canonical Schwartz space ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ and its dual ${\displaystyle {\mathcal {S}}'({\mathcal {H}}_{\pi })}$.

Define ${\displaystyle {\mathcal {G}}({\mathcal {H}}_{\pi })}$ to be the Garding domain for π. That is, ${\displaystyle {\mathcal {G}}({\mathcal {H}}_{\pi })}$ is spanned by vectors of the form

${\displaystyle \int \psi (g)\pi (g)y\,d\mu (g)}$

where ${\displaystyle y\in {\mathcal {H}}_{\pi }}$, ${\displaystyle d\mu }$ is the Haar measure on H(V), ${\displaystyle \psi \in C_{0}^{\infty }(H(V))}$ the space of smooth functions of compact support on H(V). It is known that

1. ${\displaystyle {\mathcal {G}}({\mathcal {H}}_{\pi })}$ is dense in ${\displaystyle {\mathcal {H}}_{\pi }}$
2. Qj and Pj are essentially self-adjoint on ${\displaystyle {\mathcal {G}}({\mathcal {H}}_{\pi })}$
3. Qj and Pj leave ${\displaystyle {\mathcal {G}}({\mathcal {H}}_{\pi })}$ invariant.

Denote by (Pi)0 and (Qj)0 the restrictions of Pi and Qj to to the Garding domain (where the products of these operators make sense). Then denote by QmPn the adjoint in ${\displaystyle {\mathcal {H}}_{\pi }}$ of (PnQm)0 (here multiindex notation is to be understood). Let U be the space of linear combinations of operators of the form QmPn.

Define the Schwartz space ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ by

${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })=\bigcap _{A\in U}\operatorname {Dom} (A)}$

Define the topology on ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ by a family of seminorms

${\displaystyle \|x\|_{A}=\|Ax\|}$

where ||•|| is the Hilbert space norm. Note that ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ is dense in ${\displaystyle {\mathcal {H}}_{\pi }}$ since it contains ${\displaystyle {\mathcal {G}}({\mathcal {H}}_{\pi })}$ .

Theorem
1. ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })=\{x\in {\mathcal {H}}_{\pi }:\pi (n)x\in C^{\infty }(H(V),{\mathcal {H}}_{\pi })}$
2. The family of seminorms ${\displaystyle \|x\|_{a,b}=\|Q^{a}P^{b}x\|}$ is equivalent.
3. ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ is invariant under U.

From 1, it follows that ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ does not depend on the choice of basis {e,f}. From 1 and 3, it follows that the commutation relations [Qj,Pk] = iδjk, [Pj,Pk] = 0, [Qj,Qk] = 0, and [Aj,Ak*]= δjk, [Aj,Ak]= 0, [Aj*,Ak*] = 0 hold on ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$.

For multi-index ${\displaystyle \ell }$, define an operator ${\displaystyle K_{\ell }}$ by

${\displaystyle K_{m}=\sum _{|\ell |\leq m}c_{\ell }^{m}A^{\ell }(A^{*})^{\ell }}$

where the constants can are defined by

${\displaystyle (1+|z|^{2})^{m}=\sum _{|\ell |\leq m}c_{\ell }^{m}|z_{1}|^{2\ell _{1}}\dotsb |z_{n}|^{2\ell _{n}}}$

Define the inner product on ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ by

${\displaystyle (x,y)_{B,m}=(x,K_{m}y).}$

Note that ||x|| ≤ ||x||B,m.

Theorem
The family of seminorms {||∙||a,b : a,bNN} and {||∙||B,m : mN} are equivalent.
Proof found in [1]

The first norm becomes transparent in the Schrödinger realization and the second norm in the BFS realization.

${\displaystyle {\mathcal {S}}'({\mathcal {H}}_{\pi })}$ is the space of conjugate linear continuous functionals on ${\displaystyle {\mathcal {S}}({\mathcal {H}}_{\pi })}$ and is called the space of abstract tempered distributions. It is the usual space of tempered distributions in the Schrödinger realization, and will admit a transparent description in the BFS realization as well.

The BFS realization

Polarizations

In the sequel, let the symplectic vector space and an admissible complex structure be fixed (so we will no longer carry notation for it). Let f be a nonzero element of ${\displaystyle {\mathfrak {H}}^{*}}$. Let Bf(u,v) = f([u,v]), an alternating bilinear form. One checks that ${\displaystyle \operatorname {ker} (B_{f}^{\mathbb {C} })=\mathbb {C} E.}$ (This seems to only be true if one assumed f(E) ≠ 0?)

Definition
A positive polarization at f is a subalgebra ${\displaystyle {\mathfrak {P}}}$ of ${\displaystyle {\mathfrak {H}}^{\mathbb {C} }}$ which satisfies
1. ${\displaystyle {\mathfrak {P}}}$ is a maximal totally isotropic subspace for Bf.
2. ${\displaystyle {\mathfrak {P}}+{\overline {\mathfrak {P}}}}$ is a subalgebra of ${\displaystyle {\mathfrak {H}}^{\mathbb {C} }}$
3. ${\displaystyle iB_{f}^{\mathbb {C} }(h,{\overline {h}})\geq 0}$ for all h in ${\displaystyle {\mathfrak {P}}}$

${\displaystyle {\mathfrak {P}}}$ is said to be a real polarization if ${\displaystyle {\mathfrak {P}}={\overline {\mathfrak {P}}}}$ and totally complex if ${\displaystyle {\mathfrak {P}}+{\overline {\mathfrak {P}}}={\mathfrak {H}}^{\mathbb {C} }}$

We take f to be the dual vector to E.

Example 1
The Schrödinger polarization
Let Q be a real n-dimensional subspace of VJ so that gJ is real on Q, and set P=JQ so that V = QP. Then ${\displaystyle {\mathfrak {P}}_{S}=\mathbb {C} \mathbf {P} +\mathbb {C} E}$ is a real positive polarization at E* known as the Schrödinger polarization. It will lead to the usual Schrödinger realization of H.
Example 2
The BFS polarization
${\displaystyle {\mathfrak {P}}_{B}=V_{J}^{-}+\mathbb {C} E}$ is a totally complex positive polarization at E* called the BFS polarization. It will lead to the usual Fock realization of H.

The method of holomorphically induced representations allows one to obtain an irreducible unitary representation of H using a positive polarization at E*. Recall that the action of a tangent vector vp of a Lie group is given by

${\displaystyle v_{p}(\phi )=\left.{\frac {d}{dt}}\right|_{t=0}\phi (pe^{tv}).}$

Denote by ${\displaystyle C^{\infty }({\mathfrak {P}})}$ the space of smooth complex functions on H which satisfy

${\displaystyle v(\phi )=-iE^{*}(v)\phi }$

for ${\displaystyle v\in {\mathfrak {P}}.}$

Then

References

1. ^ a b Babbit, D.: Certain Hilbert Spaces of Analytic Functions Associated With the Heisenberg Group, Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, 19-82 Princeton University Press (1976).
2. ^ Fock, V.: Verallgemeinerung und Lösung der Diracschen Statistischen Gleichung, Z. Physick, 49, 339-357 (1928).
3. ^ Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform - Part I, Comm. on Pure and Applied Math., 14, 187-214 (1961).
4. ^ Segal, I. E.: Mathematical characterization of the physical vacuum for a linear Bose-Einstein field, I11. J. Math., 6, 500-523 (1962)
5. ^ Bargmann, V.: Remarks on a Hilbert space of analytic functions, Proc. N.A.S., 48, 199-204 (1962)