25.3 The metaplectic representation and U(d) Sp(2d, R)

25.3 The metaplectic representation and U(d) Sp(2d, R)

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · unitary representation · irreducible representation

Turning to the quantization problem, we would like to extend the discussion of quantization of quadratic combinations of complex coordinates on phase space from the $d=1$ case of chapter 24 to the general case. For any $j,k$ one can take

$${\overline{z}}_j{\overline{z}}_k\to -i{a}_j{a}_k,z_j{z}_k\to -i{a}_j^{†}{a}_k^{†}$$

There is no ambiguity in the quantization of the two subalgebras given by pairs of the $z_j$ coordinates or pairs of the ${\overline{z}}_j$ coordinates since creation operators commute with each other, and annihilation operators commute with each other.

If $j\ne$ k one can quantize by taking

$$z_j{\overline{z}}_k\to -i{a}_j^{†}{a}_k=-i{a}_k{a}_j^{†}$$

and there is again no ordering ambiguity. If $j=k$ , as in the $d=1$ case there is a choice to be made. One possibility is to take

$$z_j{\overline{z}}_j\to -i\frac{1}{2}(a_j{a}_j^{†}+a_j^{†}{a}_j)=-i\left(a_j^{†}{a}_j+\frac{1}{2}\right)$$

which will have the proper sp $(2d,\mathbf{R})$ commutation relations (in particular for commutators of $a_j^2$ with $(a_j^{†}{)}^2)$ , but require going to a double cover to get a true representation of the group. The Bargmann-Fock construction thus gives us a unitary representation of $\mathfrak{u}(d)$ on Fock space ${\mathcal{F}}_d,$ but after exponentiation this is a representation not of the group $U(d)$ , but of a double cover we call $\widetilde{U(d)}$

One could instead quantize using normal ordered operators, taking

$$z_j{\overline{z}}_j\to -i{a}_j^{†}{a}_j$$

The definition of normal ordering in section 24.3 generalizes simply, since the order of annihilation and creation operators with diferent values of $j$ is immaterial. Using this normal ordered choice, the usual quantized operators of the Bargmann-Fock representation are shifted by a scalar $\frac{1}{2}$ for each $j,$ and after exponentiation the state space $\mathcal{H}={\mathcal{F}}_d$ provides a representation of $U(d)$ , with no need for a double cover. As a u(d) representation however, this does not extend to a representation of $\mathfrak{sp}(2d,\mathbf{R})$ , since commutation of $a_j^2$ with $(a_j^{†}{)}^2$ can land one on the unshifted operators.

Since the normal ordering doesn’t change the commutation relations obeyed by products of the form $a_j^{†}{a}_k,$ , the quadratic expression for ${\mu }_A$ can be quantized using normal ordering, and get quadratic combinations of the $a_j,a_k^{†}$ with the same commutation relations as in theorem 25.1. Letting

$$U_A^{\prime }=\sum _{j,k}{a}_j^{†}{A}_{jk}{a}_k\text{\hspace{1em}(25.6)}$$

we have

Theorem 25.2

For $A\in \mathfrak{gl}(d,\mathbf{C})$ a d by $d$ complex matrix

$$[U_A^{\prime },U_{A^{\prime }}^{\prime }]=U_{[A,A^{\prime }]}^{\prime }$$

As a result

$$A\in \mathfrak{gl}(d,\mathbf{C})\to U_A^{\prime }$$

is a Lie algebra representation of $\mathfrak{gl}(d,\mathbf{C})$ on $\mathcal{H}=\mathbf{C}[z_1,\dots ,z_d]$ , the harmonic oscillator state space in $d$ degrees of freedom.

In addition (for column vectors $a$ with components $a_1,\dots ,a_d)$

$$[U_A^{\prime },{\mathbf{a}}^{†}]=A^T{\mathbf{a}}^{†},[U_A^{\prime },\mathbf{a}]=-A\mathbf{a}\text{\hspace{1em}(25.7)}$$

Proof. Essentially the same proof as 25.1.

For $A\in \mathfrak{u}(d)$ the Lie algebra representation $U_A^{\prime }$ of $\mathfrak{u}(d)$ exponentiates to give a representation of $U(d)$ on $\mathcal{H}=\mathbf{C}[z_1,\dots ,z_d]$ by operators

$$U_{e^A}=e^{U_A^{\prime }}$$

These satisfy

$$U_{e^A}{\mathbf{a}}^{†}(U_{e^A}{)}^{-1}=e^{A^T}{\mathbf{a}}^{†},U_{e^A}\mathbf{a}(U_{e^A}{)}^{-1}=e^{\overline{A^T}}\mathbf{a}\text{\hspace{1em}(25.8)}$$

(the relations 25.7 are the derivative of these). This shows that the $U_{e^A}$ are intertwining operators for a $U(d)$ action on annihilation and creation operators that preserves the canonical commutation relations. Here the use of normal ordered operators means that $U_A^{\prime }$ is a representation of $\mathfrak{u}(d)$ that difers by a constant from the metaplectic representation, and $U_{e^{\iota }}$ A difers by a phase-factor. This does not afect the commutation relations with $U_A^{\prime }$ or the conjugation action of $U_{e^A}$ . The representation constructed this way difers in two ways from the metaplectic representation. It acts on the same space $\mathcal{H}={\mathcal{F}}_d$ , but it is a true representation of $U(d)$ , no double cover is needed. It also does not extend to a representation of the larger group Sp(2d, $R$).

The operators $U_A^{\prime }$ and $U_{e^A}$ commute with the Hamiltonian operator for the harmonic oscillator (the quantization of equation 25.5). For physicists this is quite useful, as it provides a decomposition of energy eigenstates into irreducible representations of $U(d)$ . For mathematicians, the quantum harmonic oscillator state space provides a construction of a large class of irreducible representations of $U(d)$ , by considering the energy eigenstates of a given energy.

Previous section: 25.2 Complex coordinates on phase space and U(d) ⊂ Sp(2d, R) · Next section: 25.4 Examples in d=2 and 3 · 中文版本

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