24.1 The metaplectic representation for d=1 in terms of a and a ^

24.1 The metaplectic representation for d=1 in terms of a and a ^

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

Poisson brackets of order two combinations of z and $\overline{z}$ can easily be computed using the basic relation $\{z,\overline{z}\}=i$ and the Leibniz rule. $\mathrm{On}$ basis elements $z^2,{\bar{z}}^2$ , zz the non-zero brackets are

$$\{z\overline{z},z^2\}=-2i{z}^2,\{z\overline{z},{\overline{z}}^2\}=2i{\overline{z}}^2,\{{\overline{z}}^2,z^2\}=-4iz\overline{z}$$

Recall from equation 16.8 that quadratic real combinations of $p$ and $q$ can be identified with the Lie algebra $\mathfrak{sl}(2,\mathbf{R})$ of traceless 2 by $2$ real matrices with basis

$$E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),G=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

Since we have complexified, allowing complex linear combinations of basis elements, our quadratic combinations of z and z are in the complexification of $\mathfrak{sl}(2,\mathbf{R})$ . This is the Lie algebra $\mathfrak{sl}(2,\mathbf{C})$ of traceless $2$ by $2$ complex matrices. We can take as a basis of $\mathfrak{sl}(2,\mathbf{C})$ over the complex numbers

$$Z=E-F,X_{\pm }=\frac{1}{2}(G\pm i(E+F))$$

which satisfy

$$[Z,X_{-}]=-2i{X}_{-},[Z,X_{+}]=2i{X}_{+},[X_{+},X_{-}]=-iZ$$

and then use as our isomorphism between quadratics in $z,\overline{z}$ and $\mathfrak{sl}(2,\mathbf{C})$

$$\frac{{\overline{z}}^2}{2}\leftrightarrow X_{+},\frac{z^2}{2}\leftrightarrow X_{-},z\overline{z}\leftrightarrow Z$$

The element

$$z\overline{z}=\frac{1}{2}(q^2+p^2)\leftrightarrow Z=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

exponentiates to give a $SO(2)=U(1)$ subgroup of $SL(2,\mathbf{R})$ with elements of the form

$$e^{\theta Z}=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)\text{\hspace{1em}(24.1)}$$

Note that $h=\frac{1}{2}(p^2+q^2)=z\overline{z}$ is the classical Hamiltonian function for the harmonic oscillator.

We can now quantize quadratics in $z$ and $\overline{z}$ using annihilation and creation operators acting on the Fock space $\mathcal{F}.$ . There is no operator ordering ambiguity for

$$z^2\to (a^{†}{)}^2=z^2,{\overline{z}}^2\to a^2=\frac{d^2}{d{z}^2}$$

For the case of $z\overline{z}(\mathrm{which}$ is real), in order to get the $\mathfrak{sl}(2,\mathbf{R})$ commutation relations to come out right (in particular, the Poisson bracket $\{{\overline{z}}^2,z^2\}=-4iz\overline{z})$ 2 we must take the symmetric combination

$$z\overline{z}\to \frac{1}{2}(a{a}^{†}+a^{†}a)=a^{†}a+\frac{1}{2}=z\frac{d}{dz}+\frac{1}{2}$$

(which of course is the standard Hamiltonian for the quantum harmonic oscillator).

Multiplying as usual by −$i$ (to get a unitary representation of the real Lie algebra $\mathfrak{sl}(2,\mathbf{R}))$ , an extension of the Bargmann-Fock representation ${\Gamma }_{BF}^{\prime }$ of h₃ $\otimes \mathbf{C}$ (see section 22.4) to an $\mathfrak{sl}(2,\mathbf{C})$ representation can be defined by taking

$${\Gamma }_{BF}^{\prime }(X_{+})=-\frac{i}{2}{a}^2,{\Gamma }_{BF}^{\prime }(X_{-})=-\frac{i}{2}(a^{†}{)}^2,{\Gamma }_{BF}^{\prime }(Z)=-i\frac{1}{2}(a^{†}a+a{a}^{†})$$

This is the right choice of ${\Gamma }_{BF}^{\prime }(Z)$ to get an $\mathfrak{sl}(2,\mathbf{C})$ representation since

$$\begin{array}{rl}[{\Gamma }_{BF}^{\prime }(X_{+}),{\Gamma }_{BF}^{\prime }(X_{-})]& =\left[-\frac{i}{2}{a}^2,-\frac{i}{2}(a^{†}{)}^2\right]=-\frac{1}{2}(a{a}^{†}+a^{†}a)\\ & =-i{\Gamma }_{BF}^{\prime }(Z)={\Gamma }_{BF}^{\prime }([X_{+},X_{-}])\end{array}$$

As a representation of the real sub-Lie algebra $\mathfrak{sl}(2,\mathbf{R})$ of $\mathfrak{sl}(2,\mathbf{C})$ , one has (using the fact that $G,E+F,E-F$ is a real basis of $\mathfrak{sl}(2,\mathbf{R}))$

Definition (Metaplectic representation of ${ \mathfrak { s l } } ( 2 , \mathbf { R } )

)$.Therepresentation$\Gamma _ { B F } ^ { \prime }$on$\mathcal { F }$ given by

$${\Gamma }_{BF}^{\prime }(G)={\Gamma }_{BF}^{\prime }(X_{+}+X_{-})=-\frac{i}{2}((a^{†}{)}^2+a^2)$$

$${\Gamma }_{BF}^{\prime }(E+F)={\Gamma }_{BF}^{\prime }(-i(X_{+}-X_{-}))=-\frac{1}{2}((a^{†}{)}^2-a^2)\text{\hspace{1em}(24.2)}$$ $${\Gamma }_{BF}^{\prime }(E-F)={\Gamma }_{BF}^{\prime }(Z)=-i\frac{1}{2}(a^{†}a+a{a}^{†})$$

is a representation of $\mathfrak{sl}(2,\mathbf{R})$ , called the metaplectic representation.

Note that this is clearly a unitary representation, since all the operators are skew-adjoint (using the fact that a and $a^{†}$ are each other’s adjoints).

This representation ${\Gamma }_{BF}^{\prime }$ on $\mathcal{F}$ will be unitarily equivalent using the Bargmann transform (see section $23.5)$ to the Schr¨odinger representation ${\Gamma }_S^{\prime }$ found earlier when quantizing $q^2,p^2,pq$ as operators on $\mathcal{H}=L^2(\mathbf{R})$ . For many purposes it is however much easier to work with since it can be studied as the state space of the quantum harmonic oscillator, which comes with a basis of eigenvectors of the number operator $a^{†}a$ . The Lie algebra acts simply on such eigenvectors by quadratic expressions in the annihilation and creation operators.

One thing that can now easily be seen is that this representation ${\Gamma }_{BF}^{\prime }$ does not integrate to give a representation of the group $SL(2,\mathbf{R})$ . If the Lie algebra representation ${\Gamma }_{BF}^{\prime }$ comes from a Lie group representation ${\Gamma }_{BF}$ of $SL(2,\mathbf{R})$ , we have

$${\Gamma }_{BF}(e^{\theta Z})=e^{\theta {\Gamma }_{BF}^{\prime }(Z)}$$

where

$${\Gamma }_{BF}^{\prime }(Z)=-i\left(a^{†}a+\frac{1}{2}\right)=-i\left(N+\frac{1}{2}\right)$$

so

$${\Gamma }_{BF}(e^{\theta Z})|n⟩=e^{-i\theta (n+\frac{1}{2})}|n⟩$$

Taking $\theta =2\pi$ , this gives an inconsistency

$${\Gamma }_{BF}(1)|n⟩=-|n⟩$$

This is the same phenomenon first described in the context of the Schr¨odinger representation in section 17.1.

As remarked there, it is the same sort of problem we found when studying the spinor representation of the Lie algebra $\mathfrak{so}(3)$ . Just as in that case, the problem indicates that we need to consider not the group $SL(2,\mathbf{R})$ , but a double cover, the metaplectic group $\mathcal{M}p(2,\mathbf{R})$ . The behavior here is quite a bit more subtle than in the $Spin(3)$ double cover case, where $Spin(3)$ was the group $SU(2)$ 2 and topologically the only non-trivial cover of $SO(3)$ was the $Spin(3)$ one since ${\pi }_1(SO(3))={\mathbf{Z}}_2$ . Here ${\pi }_1(SL(2,\mathbf{R}))=\mathbf{Z}.$ , and each extra time one goes around the $U(1)$ subgroup we are looking at, one gets a topologically diferent noncontractible loop in the group. As a result, $SL(2,\mathbf{R})$ has lots of non-trivial covering groups, of which only one interests us, the double cover $\mathcal{M}p(2,\mathbf{R})$ . In particular, there is an infinite-sheeted universal cover $SL(2,\mathbf{R})$ , but that plays no role here.

Digression. This group $\mathcal{M}p(2,\mathbf{R})$ is quite unusual in that it is a finite dimensional Lie group, but does not have any sort of description as a group of finite dimensional matrices. This is due to the fact that all its finite dimensional irreducible representations are the same as those of $SL(2,\mathbf{R})$ , which has the same Lie algebra (these are representations on homogeneous polynomials in two variables, those first studied in chapter 8, which are $SL(2,\mathbf{C})$ representations which can be restricted to $SL(2,\mathbf{R}))$ . These finite dimensional representations factor through SL(2, $R$) so their matrices don’t distinguish between two diferent elements of Mp(2, $R$) that correspond as $SL(2,\mathbf{R})$ elements.

There are no faithful finite dimensional representations of $\mathcal{M}p(2,\mathbf{R})$ itself which could be used to identify M p(2, $R$) with a group of matrices. The only faithful irreducible representation available is the infinite dimensional one we are studying. Note that the lack of a matrix description means that this is a case where the definition we gave of a Lie algebra in terms of the matrix exponential does not apply. The more general geometric definition of the Lie algebra of a group in terms of the tangent space at the identity of the group does apply, although to do this one really needs a construction of the double cover $\mathcal{M}p(2,\mathbf{R})$ which is quite non-trivial and not done here. This is not a problem for purely

Lie algebra calculations, since the Lie algebras of $\mathcal{M}p(2,\mathbf{R})$ and $SL(2,\mathbf{R})$ can be identified.

Another aspect of the metaplectic representation that is relatively easy to see in the Bargmann-Fock construction is that the state space $\mathcal{F}$ is not an irreducible representation, but is the sum of two irreducible representations

$$\mathcal{F}={\mathcal{F}}_{even}\oplus {\mathcal{F}}_{odd}$$

where ${\mathcal{F}}_{even}$ consists of the even functions of $z,{\mathcal{F}}_{odd}$ of odd functions of z. On the subspace ${\mathcal{F}}^{fin}\subset \mathcal{F}$ of finite sums of the number eigenstates, these are the even and odd degree polynomials. Since the generators of the Lie algebra representation are degree two combinations of annihilation and creation operators, they will take even functions to even functions and odd to odd. The separate irreducibility of these two pieces is due to the fact that (when n and m have the same parity), one can get from state $|n⟩$ to any another $|m⟩$ by repeated application of the Lie algebra representation operators.

Chapter contents · Next section: 24.2 Intertwining operators in terms of a and a ^ · 中文版本

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