In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized by David Mumford.
Construction The theta representation is a representation of the continuous Heisenberg group H 3 ( R ) {\displaystyle H_{3}(\mathbb {R} )} over the field of the real numbers. In this representation, the group elements act on a particular Hilbert space. The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of the isomorphism to the usual representations.
Operators and group law Let f(z) be a holomorphic function, let a and b be real numbers, and let τ {\displaystyle \tau } be an arbitrary fixed complex number in the upper half-plane; that is, so that the imaginary part of τ {\displaystyle \tau } is positive. Define the operators Sa and Tb such that they act on holomorphic functions as
( S a f ) ( z ) = f ( z + a ) = exp ( a ∂ z ) f ( z ) {\displaystyle (S_{a}f)(z)=f(z+a)=\exp(a\partial _{z})f(z)}
and
( T b f ) ( z ) = exp ( i π b 2 τ + 2 π i b z ) f ( z + b τ ) = exp ( i π b 2 τ + 2 π i b z + b τ ∂ z ) f ( z ) . {\displaystyle (T_{b}f)(z)=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z).}
It can be seen that each operator generates a one-parameter subgroup:
S a 1 ( S a 2 f ) = S a 1 + a 2 f {\displaystyle S_{a_{1}}\left(S_{a_{2}}f\right)=S_{a_{1}+a_{2}}f}
and
T b 1 ( T b 2 f ) = T b 1 + b 2 f . {\displaystyle T_{b_{1}}\left(T_{b_{2}}f\right)=T_{b_{1}+b_{2}}f.}
However, S and T do not commute:
S a T b = exp ( 2 π i a b ) T b S a . {\displaystyle S_{a}T_{b}=\exp(2\pi iab)T_{b}S_{a}.}
Thus S {\displaystyle S} and T {\displaystyle T} together with a unitary phase form a nilpotent Lie group, the continuous real Heisenberg group, parametrizable as H = U ( 1 ) × R × R {\displaystyle H=U(1)\times \mathbb {R} \times \mathbb {R} } , where U ( 1 ) {\displaystyle U(1)} is the unitary group. A general group element U τ ( λ , a , b ) ∈ H {\displaystyle U_{\tau }(\lambda ,a,b)\in H} then acts on a holomorphic function f(z) as
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