20.3 Explicit calculations

As a balance to the abstract discussion so far in this chapter, in this section we’ll work out explicitly what happens for some simple examples of subgroups of $S p (2 d, R)$ acting on phase space. They are chosen because of important later applications, but also because the calculations are quite simple, while demonstrating some of the phenomena that occur. The general story of how to explicitly construct the full metaplectic representation is quite a bit more complex. These calculations will also make clear the conventions being chosen, and show the basic structure of what the quadratic operators corresponding to actions of subgroups of the symplectic group look like, a structure that will reappear in the much more complicated infinite dimensional quantum field theory examples we will come to later.

20.3.1 The SO(2) action by rotations of the plane for d=2

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

In the case $d = 2$ one can consider the $S O (2)$ group which acts as the group of rotations of the configuration space $R^{2}$ , with a simultaneous rotation of the momentum space. This leaves invariant the Poisson bracket and so is a subgroup of $S p (4, R)$ (this is just the $S O (2)$ subgroup of $E (2)$ studied in section 19.1).

From the discussion in section 16.2, this $S O (2)$ acts by automorphisms on the Heisenberg group $H_{5}$ and Lie algebra $h_{5}$ , both of which can be identified with $M \oplus R$ , by an action leaving invariant the R component. The group $S O (2)$ acts on $c_{q_{1}} q_{1} + c_{q_{2}} q_{2} + c_{p_{1}} p_{1} + c_{p_{2}} p_{2} \in M$ by

c_{q_{1}} c_{q_{2}} c_{p_{1}} c_{p_{2}} \to g c_{q_{1}} c_{q_{2}} c_{p_{1}} c_{p_{2}} = cos θ sin θ 00 - sin θ cos θ 00 00 cos θ sin θ 00 - sin θ cos θ c_{q_{1}} c_{q_{2}} c_{p_{1}} c_{p_{2}}

so $g = e^{θ L}$ where $L \in sp (4, R)$ is given by

L = 0100 - 1 000 0001 00 - 1 0

$L$ acts on phase space coordinate functions by

q_{1} q_{2} p_{1} p_{2} \to L^{T} q_{1} q_{2} p_{1} p_{2} = q_{2} - q_{1} p_{2} - p_{1}

By equation 16.22, with

A = (01 - 1 0), B = C = 0

the quadratic function $μ_{L}$ that satisfies

⎩ ⎨ ⎧ μ_{L}, q_{1} q_{2} p_{1} p_{2} ⎭ ⎬ ⎫ = L^{T} q_{1} q_{2} p_{1} p_{2} = 0 - 1 00 1000 000 - 1 0010 q_{1} q_{2} p_{1} p_{2} = q_{2} - q_{1} p_{2} - p_{1}

μ_{L} = - q \cdot (01 - 1 0) p = q_{1} p_{2} - q_{2} p_{1}

which is just the formula for the angular momentum

l = q_{1} p_{2} - q_{2} p_{1}

in $d = 2.$

Quantization gives a representation of the Lie algebra $so (2)$ with

U_{L}^{'} = - i (Q_{1} P_{2} - Q_{2} P_{1})

satisfying

[U_{L}^{'}, (Q _{2} Q _{1})] = (- Q _{1} Q _{2}), [U_{L}^{'}, (P _{2} P _{1})] = (- P _{1} P _{2})

Exponentiating gives a representation of SO(2)

U_{e^{θ L}} = e^{- i θ (Q_{1} P_{2} - Q_{2} P_{1})}

with conjugation by $U_{e^{θ L}}$ rotating linear combinations of the $Q_{1}, Q_{2}$ (or the $P_{1}, P_{2})$ each by an angle $θ$ .

U_{e^{θ L}} (c_{q_{1}} Q_{1} + c_{q_{2}} Q_{2}) U_{e^{θ L}}^{- 1} = c_{q_{1}}^{'} Q_{1} + c_{q_{2}}^{'} Q_{2}

where

(c _{q_{2}}^{'} c _{q_{1}}^{'}) = (cos θ sin θ - sin θ cos θ) (c _{q_{2}} c _{q_{1}})

These representation operators are exactly the ones found in (section 19.1) the discussion of the representation of $E (2)$ corresponding to the quantum free particle in two dimensions. There we saw that on position space wavefunctions this is just the representation induced from rotations of the position space. It also comes from the Schr¨odinger representation, by taking a specific quadratic combination of the $Q_{j}, P_{j}$ operators, the one corresponding to the quadratic function $l$ . Note that there is no ordering ambiguity in this case since one does not multiply $q_{j}$ and $p_{j}$ with the same value of $j .$ . Also note that for this $S O (2)$ the double cover is trivial: as one goes around the circle in $S O (2)$ once, the operator $U_{e^{θ L}}$ is well-defined and returns to its initial value. As far as this subgroup of $S p (4, R)$ is concerned, there is no need to consider the double cover $M p (4, R)$ to get a well-defined representation.

The case of the group $S O (2) \subset S p (4, R)$ can be generalized to a larger subgroup, the group $G L (2, R)$ of all invertible linear transformations of $R^{2}$ performed simultaneously on position and momentum space. Replacing the matrix L by

(A 0 0 A)

for A any real 2 by 2 matrix

A = (a_{11} a_{21} a_{12} a_{22})

we get an action of the group $G L (2, R) \subset S p (4, R)$ on $M$ , and after quantization a Lie algebra representation

U_{A}^{'} = i (Q_{1} Q_{2}) (a_{11} a_{21} a_{12} a_{22}) (P _{2} P _{1})

which will satisfy

[U_{A}^{'}, (Q _{2} Q _{1})] = - A (Q _{2} Q _{1}), [U_{A}^{'}, (P _{2} P _{1})] = A^{T} (P _{2} P _{1})

Note that the action of $A$ on the momentum operators is the dual of the action on the position operators. Only in the case of an orthogonal action (the SO(2) earlier) are these the same, with $A^{T} = - A$

20.3.2 An SO(2) action on the d=1 phase space

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

Another sort of $S O (2)$ action on phase space provides a $d = 1$ example that mixes position and momentum coordinates. This will lead to quite non-trivial intertwining operators, with an action on wavefunctions that does not come about as an induced action from a group action on position space. This example will be studied in much greater detail when we get to the theory of the quantum harmonic oscillator, beginning with chapter 22. Such a physical system is periodic in time, so the usual group $R$ of time translations becomes this $S O (2)$ , with the corresponding intertwining operators giving the time evolution of the quantum states.

In this case $d = 1$ and one has elements $g \in S O (2) \subset S p (2, R)$ acting on $c_{q} q + c_{p} p \in M$ by

(c _{p} c _{q}) \to g (c _{p} c _{q}) = (cos θ - sin θ sin θ cos θ) (c _{p} c _{q})

g = e^{θ L}

where

L = (0 - 1 10)

(Note that for such phase space rotations, we are making the opposite choice for convention of the positive direction of rotation, clockwise instead of counterclockwise).

To find the intertwining operators, we first find the quadratic function $μ_{L}$ in $q, p$ that satisfies

{μ_{L}, (p q)} = L^{T} (p q) = (q - p)

By equation 16.7 this is

μ_{L} = \frac{1}{2} (q p) (1001) (p q) = \frac{1}{2} (q^{2} + p^{2})

Quantizing $μ_{L}$ using the Schr¨odinger representation $Γ_{S}^{'}$ , one has a unitary Lie algebra representation $U^{'}$ of $so (2)$ with

U_{L}^{'} = - \frac{i}{2} (Q^{2} + P^{2})

satisfying

[U_{L}^{'}, (P Q)] = (Q - P) (20.6)

and intertwining operators

U_{g} = e^{θ U_{L}^{'}} = e^{- i \frac{θ}{2} (Q^{2} + P^{2})}

These give a representation of $S O (2)$ only up to a sign, for reasons mentioned in section 17.1 that will be discussed in more detail in chapter 24.

Conjugating the Heisenberg Lie algebra representation operators by the unitary operators $U_{g}$ intertwines the representations corresponding to rotations of the phase space plane by an angle $θ$

e^{- i \frac{θ}{2} (Q^{2} + P^{2})} (P Q) e^{i \frac{θ}{2} (Q^{2} + P^{2})} = (cos θ sin θ - sin θ cos θ) (P Q) (20.7)

Note that this is a diferent calculation than in the spin case where we also constructed a double cover of $S O (2)$ . Despite the diferent context $(S O (2)$ acting on an infinite dimensional state space), again one sees an aspect of the double cover here, as either $U_{g}$ or $- U_{g}$ will give the same $S O (2)$ rotation action on the operators $Q, P$ (while each having a diferent action on the states, to be worked out in chapter 24).

In our discussion here we have blithely assumed that the operator $U_{L}^{'}$ can be exponentiated, but doing so turns out to be quite non-trivial. As remarked earlier, this representation on wavefunctions does not arise as the induced action from an action on position space. $U_{L}^{'}$ is (up to a factor of i) the Hamiltonian operator for a quantum system that is not translation invariant. It involves quadratic operators in both $Q$ and $P,$ so neither the position space nor momentum space version of the Schr¨odinger representation can be used to make the operator a multiplication operator. Further details of the construction of the needed exponentiated operators will be given in section 23.4.

20.3.3 The Fourier transform as an intertwining operator

For another indication of the non-trivial nature of the intertwining operators of section 20.3.2, note that a group element $g$ acting by $a \frac{π}{2}$ rotation of the $d = 1$ phase space interchanges the role of $q$ and $p .$ It turns out that the corresponding intertwining operator $U_{g}$ is closely related to the Fourier transform $F .$ . Up to a phase factor $e^{i \frac{π}{4}}$ , Fourier transformation is just such an intertwining operator: we will see in section 23.4 that, acting on wavefunctions,

U_{e^{\frac{π}{2} L}} = e^{i \frac{π}{4}} F

Squaring this $gi$ ves

U_{e^{π L}} = (U_{e^{\frac{π}{2} L}})^{2} = i F^{2}

and we know from the definition of $F$ and Fourier inversion that

F^{2} ψ (q) = ψ (- q)

The non-trivial double cover here appears because

U_{e^{2 π L}} = - F^{4} = - 1

which takes a wavefunction $ψ (q) to - ψ (q)$

20.3.4 An R action on the d=1 phase space

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue

For another sort of example in $d = 1$ , consider the action of a subgroup $R \subset$ $S L (2, R)$ on $d = 1$ phase space by

(c _{p} c _{q}) \to g (c _{p} c _{q}) = (e^{r} 0 0 e^{- r}) (c _{p} c _{q})

where

g = e^{r L}, L = (10 0 - 1)

Now, by equation 16.7 the moment map will be

μ_{L} = \frac{1}{2} (q p) (0 - 1 - 1 0) (p q) = - q p

which satisfies

{μ_{L}, (p q)} = (- p q)

Quantization gives intertwining operators by

U_{L}^{'} = - \frac{i}{2} (QP + P Q), U_{g} = e^{r U_{L}^{'}} = e^{- \frac{i r}{2} (QP + P Q)}

These act on operators $Q$ and P by a simple rescaling

e^{- \frac{i r}{2} (QP + P Q)} (P Q) e^{\frac{i r}{2} (QP + P Q)} = (e^{r} 0 0 e^{- r}) (P Q)

Note that in the Schr¨odinger representation

- i \frac{1}{2} (QP + P Q) = - i (QP - \frac{i}{2}) 1 = - q \frac{d}{d q} - \frac{1}{2} 1

The operator will have as eigenfunctions

ψ (q) = q^{c}

with eigenvalues $- c - \frac{1}{2}$ . Such states are far from square-integrable, but do have an interpretation as distributions on the Schwartz space.

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20.3 Explicit calculations