38.1 Unitary transformations on H₁

38.1 Unitary transformations on H₁

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · unitary group · unitary representation

The single-particle state space ${\mathcal{H}}_1$ of non-relativistic quantum field theory can be parametrized by either wavefunctions $\psi (x)$ or their Fourier transforms $\psi (p)$ 2 and carries a Hermitian inner product

$$⟨{\psi }_1,{\psi }_2⟩=\int \overline{{\psi }_1(x)}{\psi }_2(x)dx=\int \overline{{\tilde{\psi }}_1(p)}{\tilde{\psi }}_2(p)dp$$

As a dual phase space, the symplectic structure is given by the imaginary part of this

$$\Omega ({\psi }_1,{\psi }_2)=\frac{1}{2i}\int (\overline{{\psi }_1(x)}{\psi }_2(x)-\overline{{\psi }_2(x)}{\psi }_1(x))dx$$

There is an infinite dimensional symplectic group that acts on ${\mathcal{H}}_1$ by linear transformations that preserve Ω. It has an infinite dimensional unitary subgroup, those transformations preserving the full inner product. In this chapter we’ll consider various finite dimensional groups $G$ that are subgroups of this unitary group, and see how they are represented on the quantum field theory state space. Note that there are also groups that act as symplectic but not unitary transformations of ${\mathcal{H}}_1$ , after quantization acting by a unitary representation on the multi-particle state space. Such actions change particle number, and the vacuum state |0⟩ in particular will not be invariant. For some indications of what happen in this more general situation, see sections 25.5 and 39.4.

The finite dimensional version of the case of unitary transformations of ${\mathcal{H}}_1$ was discussed in detail in sections 25.2 and 25.3 where we saw that the moment map for the $U(d)$ action was given by

$${\mu }_A=i\sum _{j,k}{z}_j{A}_{jk}{\overline{z}}_k$$

for A a skew-adjoint matrix. Quantization took this quadratic function on phase space to the quadratic combination of annihilation and creation operators

$$\sum _{j,k}{a}_j^{†}{A}_{jk}{a}_k$$

and exponentiation of these operators gave the unitary representation on state space.

In the quantum field theory case with dual phase space ${\mathcal{H}}_1$ , the generalization of the finite dimensional case will be

$$\mathrm{index}j\to x\mathrm{or}p$$ $$z_j\to \Psi (x)\text{ or }\alpha (p),{\overline{z}}_j\to \overline{\Psi }(x)\text{ or }\overline{\alpha }(p)$$ $$a_j^{†}\to {\hat{\Psi }}^{†}(x)\text{or}{a}^{†}(p),a_j\to \hat{\Psi }(x)\text{or}a(p)$$

The quadratic functions we will consider will be ${\mathrm{\Delta }}^6\mathrm{loca}{\mathrm{l}}^9$ , multiplying elements parametrized by the same points in position space. Often these will be diferential operators. As a result, the generalization from the finite dimensional case will take

$$\sum _{j,k}\to \int dx\mathrm{or}\int dp$$

and

$$A_{jk}\to \mathcal{O}(x)\text{ or }\mathcal{O}(p)$$

Chapter contents · Next section: 38.2 Internal symmetries · 中文版本

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