34.3 The E(3) -invariant inner product

34.3 The E(3) -invariant inner product

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · irreducible representation

One can parametrize solutions to the Pauli equation and write an $\widetilde{E(3)}$ -invariant inner product on the space of solutions in several diferent ways. Three diferent parametrizations of solutions that can be considered are:

• Using the initial data at a fixed time

$$\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)$$

Here the $\widetilde{E(3)}$ -invariant inner product is

$$⟨\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right),\left(\genfrac{}{}{0ex}{}{{\psi }_1^{\prime }(\mathbf{q})}{{\psi }_2^{\prime }(\mathbf{q})}\right)⟩={\int }_{{\mathbf{R}}^3}{\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)}^{†}\left(\genfrac{}{}{0ex}{}{{\psi }_1^{\prime }(\mathbf{q})}{{\psi }_2^{\prime }(\mathbf{q})}\right)d^3\mathbf{q}$$

This parametrization does not make visible the decomposition into irreducible representations of $\widetilde{E(3)}$

• Using the Fourier transforms

$$\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1(\mathbf{p})}{{\tilde{\psi }}_2(\mathbf{p})}\right)$$

to parametrize solutions, the invariant inner product is

$$⟨\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1(\mathbf{p})}{{\tilde{\psi }}_2(\mathbf{p})}\right),\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1^{\prime }(\mathbf{p})}{{\tilde{\psi }}_2^{\prime }(\mathbf{p})}\right)⟩={\int }_{{\mathbf{R}}^3}{\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1(\mathbf{p})}{{\tilde{\psi }}_2(\mathbf{p})}\right)}^{†}\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1^{\prime }(\mathbf{p})}{{\tilde{\psi }}_2^{\prime }(\mathbf{p})}\right)d^3\mathbf{p}$$

The decomposition of equation 34.5 can be used to express solutions of energy $E$ in terms of two-component functions ${\psi }_E(\mathbf{p})$ , with an invariant inner product on the space of such solutions given by

$$⟨{\tilde{\psi }}_E(\mathbf{p}),{\widetilde{{\psi }^{\prime }}}_E(\mathbf{p})⟩=\frac{1}{4\pi }{\int }_{S^2}{\tilde{\psi }}_E(\mathbf{p}{)}^{†}{\tilde{\psi }}_E^{\prime }(\mathbf{p})\sin (\varphi )d\varphi d\theta$$

where $(p,\varphi ,\theta )$ are spherical coordinates on momentum space and $S^2$ is the sphere of radius $\sqrt{2mE}$

The ${\tilde{\psi }}_E(\mathbf{p})$ parametrize not a single irreducible representation of $\widetilde{E(3)}$ but two of them, including both helicities.

• The spin polarization vectors and equation 34.9 can be used to parametrize solutions of fixed energy $E$ in terms of two functions ${\alpha }_{E,+}(\mathbf{p}),{\alpha }_{E,+}(\mathbf{p})$ on the sphere $|\mathbf{p}{|}^2=2mE$ . This gives an explicit decomposition into irreducible representations of $\widetilde{E(3)}$ , with the representation space a space of complex functions on the sphere, with invariant inner product for each helicity choice given by

$$⟨{\alpha }_{E,\pm }(\mathbf{p}),{\alpha }_{E,\pm }^{\prime }(\mathbf{p})⟩=\frac{1}{4\pi }{\int }_{S^2}{\alpha }_{E,\pm }(\mathbf{p}{)}^{†}{\alpha }_{E,\pm }^{\prime }(\mathbf{p})\sin (\varphi )d\varphi d\theta$$

In each case the space of solutions is a complex vector space and one can imagine trying to take it as a phase space (with the imaginary part of the Hermitian inner product providing the symplectic structure) and quantizing. This will be an example of a quantum field theory, one discussed in more detail in section 38.3.3.

Previous section: 34.2 Solutions of the Pauli equation and representations of E(3) · Next section: 34.4 The Dirac operator · 中文版本

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原书 PDF · 印刷页 358、359、360、361、362、363、364、365、366、367

来源版本：2025-10-20

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