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Proposition 16.44 If \( V \) is a finite-dimensional Hilbert space over \( \mathbb{C} \), then \( \mathrm{{PU}}\left( V\right) \) is isomorphic to a matrix Lie group. | Proof. If \( \dim V = N \), then \( \operatorname{gl}\left( V\right) \), the space of all linear maps of \( V \) to \( V \) , has dimension \( {N}^{2} \) . Given \( U \in \mathrm{U}\left( V\right) \), we can define\n\n\[ \n{C}_{U} : \operatorname{gl}\left( V\right) \rightarrow \operatorname{gl}\left( V\right) \n\]\n\nb... | No |
Proposition 16.46 Let \( \Pi : G \rightarrow \mathrm{{PU}}\left( V\right) \) be a finite-dimensional projective unitary representation of a matrix Lie group \( G \), and let \( \pi : \mathfrak{g} \rightarrow \operatorname{pu}\left( V\right) \) be the associated Lie algebra homomorphism. Then there exists a Lie algebra ... | Proof. Recall that \( \mathfrak{{pu}}\left( V\right) \cong \mathfrak{u}\left( V\right) /\{ {iaI}\} \) . That is, for each \( X \in \mathfrak{g},\pi \left( X\right) \) denotes a whole family of operator that differ by adding \( {iaI} \) . If \( Y \in \mathfrak{u}\left( n\right) \) is any representative of \( \pi \left( ... | Yes |
Theorem 16.47 Suppose \( G \) is a matrix Lie group and \( \widetilde{G} \) is a universal cover of \( G \), with covering map \( \Phi \) . Then the following hold.\n\n1. Let \( \Pi : G \rightarrow \mathrm{{PU}}\left( V\right) \) be a finite-dimensional projective unitary representation of \( G \) . Then there is an or... | Proof. If \( \mathfrak{g} \) is the Lie algebra of \( G \), Proposition 16.46 tells us that we can find an ordinary representation \( \sigma : \mathfrak{g} \rightarrow \mathfrak{u}\left( V\right) \) such that \( q \circ \sigma = \pi \) . We then define a representation \( \widetilde{\sigma } : \widetilde{\mathfrak{g}} ... | Yes |
Proposition 16.51 Every finite-dimensional unitary representation of a group or Lie algebra is completely reducible. | Proof. Suppose \( \left( {\Pi, V}\right) \) is a unitary representation of a matrix Lie group \( G \) . If \( W \) is a subspace of \( V \) invariant under each \( \Pi \left( A\right) \), then \( {W}^{ \bot } \) is invariant under each \( \Pi {\left( A\right) }^{ * } \), as the reader may easily verify. But since \( \P... | Yes |
Proposition 16.52 Suppose \( K \) is a compact matrix Lie group. For any finite-dimensional representation \( \left( {\Pi, V}\right) \) of \( K \), there exists an inner product on \( V \) such that \( \Pi \left( A\right) \) is unitary for all \( A \in G \) . In particular, every finite-dimensional representation of \(... | See Proposition 4.36 in [21]. | No |
Proposition 16.54 Suppose \( G \) is a matrix Lie group and \( \Pi : G \rightarrow \mathrm{U}\left( \mathbf{H}\right) \) is a unitary representation of \( G \) . For each \( X \in \mathfrak{g} \), let \( \pi \left( X\right) \) denote the operator in (16.5). Suppose \( V \subset \mathbf{H} \) is a finite-dimensional sub... | Proof. Since \( V \) is invariant under both \( \Pi \left( A\right) \) and \( \Pi {\left( A\right) }^{ * } = \Pi \left( {A}^{-1}\right) \), the restriction to \( V \) of each \( \Pi \left( A\right) \) is unitary. The operators \( {\left. \Pi \left( A\right) \right| }_{V} \) form a finite-dimensional unitary representat... | Yes |
For all \( \left( {a, b}\right) \in {\mathbb{R}}^{2} \), define an operator \( {T}_{\left( a, b\right) } \) on \( {L}^{2}\left( \mathbb{R}\right) \) by\n\n\[ \left( {{T}_{\left( a, b\right) }\psi }\right) \left( x\right) = {e}^{iax}\psi \left( {x - b}\right) . \]\n\nThen \( {T}_{\left( a, b\right) } \) is unitary for a... | The map \( \left( {a, b}\right) \rightarrow {T}_{\left( a, b\right) } \) is easily seen to be strongly continuous, and thus the map \( \left( {a, b}\right) \mapsto \left\lbrack {T}_{\left( a, b\right) }\right\rbrack \) is continuous in the sense of Definition 16.55. If a homomorphism \( S \) with the indicated properti... | Yes |
Proposition 17.2 For each \( R \in \mathrm{{SO}}\left( 3\right) \), the map \( \Pi \left( R\right) : {L}^{2}\left( {\mathbb{R}}^{3}\right) \rightarrow {L}^{2}\left( {\mathbb{R}}^{3}\right) \) is unitary. Furthermore, the map \( \Pi : \mathrm{{SO}}\left( 3\right) \rightarrow \mathrm{U}\left( {{L}^{2}\left( {\mathbb{R}}^... | Proof. Since the Lebesgue measure on \( {\mathbb{R}}^{3} \) is invariant under rotations, \( \Pi \left( R\right) \) is unitary for all \( R \in \mathrm{{SO}}\left( 3\right) \) . It is easily checked that \( \Pi \left( {{R}_{1}{R}_{2}}\right) = \) \( \Pi \left( {R}_{1}\right) \Pi \left( {R}_{2}\right) \) ; for this to b... | Yes |
Proposition 17.3 For each \( X \in \operatorname{so}\left( 3\right) \), let \( \pi \left( X\right) \) denote the skew-selfadjoint operator such that\n\n\[ \Pi \left( {e}^{tX}\right) = {e}^{{t\pi }\left( X\right) } \]\n\n(17.5)\n\nThen the domain of each \( \pi \left( {F}_{j}\right) \) contains the Schwartz space \( \ma... | Proof. In the case of \( {\widehat{J}}_{3} \), we compute as in Example 16.16 that \( {e}^{t{F}_{3}} \) is a counterclockwise rotation in the \( \left( {{x}_{1},{x}_{2}}\right) \) -plane. If \( \psi \) belongs to \( \mathcal{S}\left( {\mathbb{R}}^{3}\right) \) then the limit defining the derivative in (17.2) is easily ... | No |
Proposition 17.7 Let \( \pi : \operatorname{so}\left( 3\right) \rightarrow \mathrm{{gl}}\left( V\right) \) be an irreducible representation of \( \operatorname{so}\left( 3\right) \) . Then there exists an inner product on \( V \), unique up to multiplication by a constant, such that \( \pi \left( X\right) \) is skew-se... | Proof. Recalling how the operators \( {L}_{3},{L}^{ + } \), and \( {L}^{ - } \) are defined, we can see that the assertion that each \( \pi \left( X\right), X \in \mathfrak{{so}}\left( 3\right) \), is skew-self-adjoint is equivalent to the assertion that \( {L}_{3} \) is self-adjoint and that \( {L}^{ + } \) and \( {L}... | Yes |
Proposition 17.8 Suppose \( \\left( {\\pi, V}\\right) \) is an irreducible representation of \( \\mathfrak{{so}}\\left( 3\\right) \) of dimension \( {2l} + 1 \) . Define the Casimir operator \( {C}_{\\pi } \\in \\operatorname{End}\\left( V\\right) \) by the formula\n\n\[ \n{C}_{\\pi } = \\pi {\\left( {F}_{1}\\right) }^... | Proof. See Exercise 3. ∎ | No |
Proposition 17.9 Let \( \left( {\pi, V}\right) \) be any finite-dimensional representation of \( \mathfrak{{so}}\left( 3\right) \), not necessarily irreducible. Suppose \( {v}_{0} \) is a nonzero element of \( V \) such that \( {L}^{ + }{v}_{0} = 0 \) and \( {L}_{3}{v}_{0} = \lambda {v}_{0} \) for some \( \lambda \in \... | \[ {v}_{j} = {\left( {L}^{ - }\right) }^{j}{v}_{0},\;j = 0,1,\ldots ,{2l}, \] span an irreducible invariant subspace of \( V \) of dimension \( {2l} + 1 \), and \( {L}^{ + } \) , \( {L}^{ - } \), and \( {L}_{3} \) act on these vectors according to the formulas in Theorem 17.4. | Yes |
Proposition 17.10 Let \( {\pi }_{l} : \operatorname{so}\left( 3\right) \rightarrow \mathrm{{gl}}\left( V\right) \) be an irreducible representation of \( \mathfrak{{so}}\left( 3\right) \), with spin \( l \mathrel{\text{:=}} \frac{1}{2}\left( {\dim V - 1}\right) \) . If \( l \) is an integer (i.e., if the dimension of \... | Proof. If \( l \) is a half-integer, then \( {L}_{3} \) is diagonal in the basis \( \left\{ {v}_{j}\right\} \), with eigenvalues being half-integers. Thus,\n\n\[ \n{e}^{{2\pi }{\pi }_{l}\left( {F}_{3}\right) } = {e}^{{2\pi i}{L}_{3}} = - I. \n\]\n\n(Here the \ | Yes |
Lemma 17.13 Let \( \mathcal{P} \) denote the space of polynomials on \( {\mathbb{R}}^{3} \) with complex coefficients. There exists an inner product \( \langle \cdot , \cdot \rangle \) on \( \mathcal{P} \) with the property that\n\n\[ \langle p,{\Delta q}{\rangle }_{\mathcal{P}} = {\left\langle {x}^{2}p, q\right\rangle... | Proof. Although it is possible to give a combinatorial construction of the desired inner product, we can also give an analytic construction. Every polynomial \( p \) on \( {\mathbb{R}}^{3} \) certainly has a holomorphic extension to \( {\mathbb{C}}^{3} \), denoted \( {p}_{\mathbb{C}} \) . We may define, then,\n\n\[ \la... | Yes |
If \( {\mathcal{P}}_{l} \) denotes the space of polynomials on \( {\mathbb{R}}^{3} \) that are homogeneous of degree \( l \), then the Laplacian \( \Delta \) maps \( {\mathcal{P}}_{l} \) onto \( {\mathcal{P}}_{l - 2} \) for all \( l \geq 2 \) . | Proof. Let us equip the finite-dimensional spaces \( {\mathcal{P}}_{l} \) and \( {\mathcal{P}}_{l - 2} \) with the inner product from Lemma 17.13. It is easy to see that the statement, \ | No |
Corollary 17.15 Let \( l \) be a non-negative integer and let \( k = l/2 \) if \( l \) is even and let \( k = \left( {l - 1}\right) /2 \) if \( l \) is odd. Then each \( p \in {\mathcal{P}}_{l} \) can be decomposed in the form\n\n\[ p\left( \mathbf{x}\right) = {p}_{0}\left( \mathbf{x}\right) + {\left| \mathbf{x}\right|... | Proof. We proceed by induction on \( l \) . If \( l = 0 \) or \( l = 1 \), then all \( p \in {\mathcal{P}}_{l} \) are harmonic and the desired decomposition is simply \( p = {p}_{0} \) . Consider, then, some \( l \geq 2 \) and assume the result holds for all degrees less than \( l \) . Lemma 17.13 tells us that \( {\ma... | Yes |
Lemma 17.16 As in Theorem 17.4, let \( {L}_{3} = {i\pi }\left( {F}_{3}\right) = {\widetilde{J}}_{3} \) and let \( {L}^{ + } = \) \( {i\pi }\left( {F}_{1}\right) - \pi \left( {F}_{2}\right) = {\widetilde{J}}_{1} + i{\widetilde{J}}_{2} \) . For any non-negative integer \( l \), the polynomial \( p\left( {{x}_{1},{x}_{2},... | Proof. Since it is independent of \( {x}_{3} \) and holomorphic as a function of \( z \mathrel{\text{:=}} {x}_{1} + i{x}_{2} \), the polynomial \( p \) is automatically harmonic, which can also be verified by direct calculation. Meanwhile, applying \( {L}_{3} \) to \( p \) gives\n\n\[ \n- i\left( {{x}_{1}\frac{\partial... | Yes |
Corollary 17.17 The space \( {V}_{l} \) is irreducible under the action of \( \mathrm{{SO}}\left( 3\right) \) . | Proof. By Proposition 17.9, if we apply \( {L}^{ - } \) repeatedly to the polynomial \( p \), we obtain a \ | No |
Proposition 17.19 Every space of the form \( {V}_{l, f} \subset {L}^{2}\left( {\mathbb{R}}^{3}\right) \) is invariant and irreducible under the action of \( \mathrm{{SO}}\left( 3\right) \) . Conversely, every finite-dimensional, irreducible, \( \mathrm{{SO}}\left( 3\right) \) -invariant subspace of \( {L}^{2}\left( {\m... | Proof. Since the factor \( f\left( \left| \mathbf{x}\right| \right) \) is invariant under rotations, the action of \( \mathrm{{SO}}\left( 3\right) \) only affects the function \( p \) . Thus, \( {V}_{l, f} \) is isomorphic, as a representation of \( \mathrm{{SO}}\left( 3\right) \), to the space \( {V}_{l} \), which is ... | Yes |
Proposition 17.23 For any \( j = 0,1/2,1,\ldots \), let \( {V}_{j} \) denote the unique irreducible representation of \( \operatorname{so}\left( 3\right) \) of dimension \( {2j} + 1 \) . Then for any \( l \) and \( m \) with \( l \geq m \), we have\n\n\[ \n{V}_{l} \otimes {V}_{m} \cong {V}_{l + m} \oplus {V}_{l + m - 1... | The proof of this result is similar to that of Proposition 17.22, and is omitted; see Theorem D. 1 in Appendix D of [21]. An important property of this decomposition is that each irreducible representation that occurs on the right-hand side of (17.20) occurs only once. This property of the representations of \( \operat... | No |
Proposition 17.25 Let \( \mathbf{j}\left( {\mathbf{x},\mathbf{p}}\right) = \mathbf{x} \times \mathbf{p} \) denote the angular momentum function on \( {\mathbb{R}}^{3} \times {\mathbb{R}}^{3} \) . Suppose a smooth function \( \mathbf{c} : {\mathbb{R}}^{3} \times {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) transform... | Proof. Let \( R\left( \theta \right) \) denote a counterclockwise rotation by angle \( \theta \) in the \( \left( {{x}_{1},{x}_{2}}\right) \) -plane. Applying (17.21) with \( R = R\left( \theta \right) \) and looking only at the first component of the vectors, we have\n\n\[ {c}_{1}\left( {R\left( \theta \right) \mathbf... | Yes |
Proposition 17.27 If \( \\mathbf{C} \) is a vector operator, then the components of \( \\mathbf{C} \) satisfy\n\n\[ \n\\frac{1}{i\\hslash }\\left\\lbrack {{C}_{j},{\\widehat{J}}_{j}}\\right\\rbrack = 0 \n\]\n\n\( \\left( {17.27}\\right) \)\n\nfor \( j = 1,2,3 \) . Furthermore, we have\n\n\[ \n\\frac{1}{i\\hslash }\\lef... | Proof. As in the proof of Proposition 17.25, \( R\\left( \\theta \\right) \) denote a rotation in the \( \\left( {{x}_{1},{x}_{2}}\\right) \)-plane, and let \( {\\mathbf{e}}_{1} = \\left( {1,0,0}\\right) \). Applying (17.26) with \( R = R\\left( \\theta \\right) \) and \( \\mathbf{v} = {\\mathbf{e}}_{1} \), we have\n\n... | Yes |
Proposition 18.1 Let \( p \) be a harmonic polynomial on \( {\mathbb{R}}^{3} \) that is homogeneous of degree \( l \) and let \( f \) be a smooth function on \( \left( {0,\infty }\right) \) . Let \( \psi \) be the function on \( {\mathbb{R}}^{3} \smallsetminus \{ 0\} \) given by\n\n\[ \psi \left( \mathbf{x}\right) = p\... | Proof. We begin with the case \( l = 0 \), so that \( p \) is a constant - which we take to be 1-and \( \psi \) is just the radial function \( f\left( \left| \mathbf{x}\right| \right) \) . Then\n\n\[ \frac{\partial }{\partial {x}_{j}}f\left( \left| \mathbf{x}\right| \right) = \frac{df}{dr}\frac{d}{d{x}_{j}}\sqrt{{x}_{1... | Yes |
Proposition 18.2 Suppose \( p \in {V}_{l} \) and \( f \) is a smooth function on \( \left( {0,\infty }\right) \) , and let \( \psi \) by the function on \( {\mathbb{R}}^{3} \smallsetminus \{ 0\} \) given by\n\n\[ \psi \left( \mathbf{x}\right) = p\left( \frac{\mathbf{x}}{\left| \mathbf{x}\right| }\right) g\left( \left| ... | Proof. Since \( p \) is homogeneous of degree \( l \) ,\n\n\[ p\left( \frac{\mathbf{x}}{\left| \mathbf{x}\right| }\right) = \frac{p\left( \mathbf{x}\right) }{{\left| \mathbf{x}\right| }^{l}} \]\n\nThus,\n\n\[ \psi \left( \mathbf{x}\right) = p\left( \mathbf{x}\right) \left( \frac{f\left( \left| \mathbf{x}\right| \right)... | No |
Theorem 18.3 For each positive integer \( n \), let\n\n\[ \n{E}_{n} = - \frac{\mu {Q}^{4}}{2{\hslash }^{2}}\frac{1}{{n}^{2}} \]\n\nwhere \( Q \) is the charge of the electron and \( \mu \) is the reduced mass of the electron-proton system, and let\n\n\[ \n{\rho }_{n}\left( \mathbf{x}\right) = \frac{\sqrt{{8\mu }\left| ... | Proof. If \( E \) is a negative real number, we look for solutions to \( \widehat{H}\psi = {E\psi } \) of the form \( q\left( \mathbf{x}\right) f\left( \left| \mathbf{x}\right| \right) \), where \( q \in {V}_{l} \). Provided that \( f\left( r\right) \) and \( {f}^{\prime }\left( r\right) \) are bounded near the origin,... | Yes |
Corollary 18.5 Each eigenvalue \( {E}_{n} \), as given in Theorem 18.3, has multiplicity \( {n}^{2} \) . | Proof. According to Theorem 18.4, the eigenvectors in Theorem 18.3 constitute all of the eigenvectors for \( \widehat{H} \) with eigenvalue \( {E}_{n} \) . The number of independent eigenvectors with eigenvalue \( {E}_{n} \) is thus the sum of the dimensions of the spaces \( {V}_{l} \) of spherical harmonics, with \( l... | Yes |
Lemma 18.7 The Runge-Lenz vector \( \mathbf{A} \) and the Hamiltonian \( H \) in (18.17) satisfy the following Poisson bracket relations:\n\n\[ \left\{ {{A}_{j}, H}\right\} = 0 \]\n\n\[ \left\{ {{A}_{j},{A}_{m}}\right\} = - \frac{2}{\mu {Q}^{4}}{\varepsilon }_{jml}{J}_{l}H \] | We have already shown that the Runge-Lenz vector is a conserved quantity (Proposition 2.34), which is equivalent (Proposition 2.25) to saying that the Poisson bracket of \( {A}_{j} \) with \( H \) is zero, as claimed. The proof of (18.18) is deferred to Sect. 18.6. | No |
Lemma 18.12 The quantum Runge-Lenz vector \( \widehat{\mathbf{A}} \) and the Hamiltonian \( \widehat{H} \) satisfy the following commutation relations: | \[ \frac{1}{i\hslash }\left\lbrack {{\widehat{A}}_{j,}\widehat{H}}\right\rbrack = 0 \] \[ \frac{1}{i\hslash }\left\lbrack {{\widehat{A}}_{j},{\widehat{A}}_{m}}\right\rbrack = - \frac{2}{\mu {Q}^{4}}{\varepsilon }_{jml}{\widehat{J}}_{l}\widehat{H} \] (18.21) Note that since \( \widehat{H} \) commutes with rotations, it ... | Yes |
Proposition 18.15 Suppose \( {V}_{k} \) and \( {V}_{l} \) are irreducible representations of so(3) of dimensions \( {2k} + 1 \) and \( {2l} + 1 \), respectively. Then \( {V}_{k} \otimes {V}_{l} \) is irreducible when viewed as a representation of \( \mathsf{{so}}\left( 3\right) \oplus \mathsf{{so}}\left( 3\right) \) as... | Proof. To classify the irreducible representations of \( \operatorname{so}\left( 3\right) \oplus \operatorname{so}\left( 3\right) \), we could appeal to the general theory of representations of direct sums of Lie algebras. It is not hard, however, to give a direct proof using the same sort of reasoning we used in the c... | No |
Corollary 18.17 If \( n, k \), and \( {W}^{\left( n\right) } \) are as in Theorem 18.16, then for all \( \psi \in {W}^{\left( n\right) } \), we have\n\n\[ \widehat{\mathbf{I}} \cdot \widehat{\mathbf{I}}\psi = \widehat{\mathbf{J}} \cdot \widehat{\mathbf{J}}\psi = {\hslash }^{2}k\left( {k + 1}\right) .\n\] | Proof of Corollary 18.17. It is easily seen that the operators \( \widehat{\mathbf{I}} \cdot \widehat{\mathbf{I}} \) and \( \widehat{\mathbf{K}} \cdot \widehat{\mathbf{K}} \), when restricted to an irreducible subspace for the action of \( \operatorname{so}\left( 3\right) \oplus \) so(3), are equal to \( - {\hslash }^{... | Yes |
Lemma 18.18 The \( \varepsilon \) -function in Definition 18.6 satisfies the relations\n\n\[ \n{\varepsilon }_{jkl}{\varepsilon }_{jmn} = {\delta }_{km}{\delta }_{ln} - {\delta }_{kn}{\delta }_{lm} \]\n\n\[ \n{\varepsilon }_{jkl}{\varepsilon }_{jkm} = 2{\delta }_{lm} \]\n | The proof of these results is not difficult and is left to the reader (Exercise 6). | No |
Lemma 18.19 If \( \\mathbf{C},\\mathbf{D} \), and \( \\mathbf{E} \) are arbitrary vector operators, we have\n\n\\[ \n\\mathbf{C} \\cdot \\left( {\\mathbf{D} \\times \\mathbf{E}}\\right) = \\left( {\\mathbf{C} \\times \\mathbf{D}}\\right) \\cdot \\mathbf{E}\n\\]\n | Proof. The right-hand side of (18.27) is computed as \( {\\varepsilon }_{jkl}{C}_{k}{D}_{l}{E}_{j} \) . If we note that \( {\\varepsilon }_{jkl} = {\\varepsilon }_{klj} \) and then relabel the indices, we obtain \( {\\varepsilon }_{jkl}{C}_{j}{D}_{k}{E}_{l} \) , which is equal to the left-hand side of (18.27). | Yes |
Lemma 18.21 For all \( j \) and \( m \), we have\n\n\[ \left\lbrack {{\left( \mathbf{P} \times \widehat{\mathbf{J}}\right) }_{j},{\left( \mathbf{P} \times \widehat{\mathbf{J}}\right) }_{m}}\right\rbrack = - i\hslash \left( {\mathbf{P} \cdot \mathbf{P}}\right) {\varepsilon }_{jml}{\widehat{J}}_{l} \] | Proof. In computing \( \left\lbrack {{P}_{k}{\widehat{J}}_{l},{P}_{n}{\widehat{J}}_{o}}\right\rbrack \), we use repeatedly the product rule for commutators (Point 3 of Proposition 3.15). We obtain four terms, one of which is zero (the term involving \( \left\lbrack {{P}_{k},{P}_{n}}\right\rbrack \) ). We use Propositio... | Yes |
Proposition 19.10 For any unit vector \( \psi \in \mathbf{H} \), let \( \left| {\psi \rangle \langle \psi }\right| \), in accordance with Notation 3.29, denote the orthogonal projection onto the span of \( \psi \) . Then \( \left| {\psi \rangle \langle \psi }\right| \) is a density matrix and for all \( A \in \mathcal{... | Proof. Since it is an orthogonal projection, \( \left| {\psi \rangle \langle \psi }\right| \) is bounded, self-adjoint, and non-negative. To compute its trace, we choose an orthonormal basis\n\n\( \left\{ {e}_{j}\right\} \) for \( \mathbf{H} \) with \( {e}_{1} = \psi \), which gives \( \operatorname{trace}\left( \left|... | Yes |
Proposition 19.12 Suppose that \( \left( {{X}_{1},{\mu }_{1}}\right) \) and \( \left( {{X}_{2},{\mu }_{2}}\right) \) are \( \sigma \) -finite measure spaces. Then there is a unique unitary map\n\n\[ p : {L}^{2}\left( {{X}_{1},{\mu }_{1}}\right) \widehat{ \otimes }{L}^{2}\left( {{X}_{2},{\mu }_{2}}\right) \rightarrow {L... | Proof. For simplicity of notation, we suppress the dependence of \( {L}^{2} \) spaces on the measure, writing, say, \( {L}^{2}\left( {X}_{1}\right) \) rather than \( {L}^{2}\left( {{X}_{1},{\mu }_{1}}\right) \) . Consider first the algebraic (i.e., uncompleted) tensor product \( {L}^{2}\left( {X}_{1}\right) \otimes {L}... | Yes |
Lemma 19.14 For any sequence \( {A}_{n} \in \mathcal{B}\left( {\mathbf{H}}_{1}\right) \), if \( \begin{Vmatrix}{{A}_{n}\psi - {A\psi }}\end{Vmatrix} \rightarrow 0 \) for some \( A \in \mathcal{B}\left( \mathbf{H}\right) \) and all \( \psi \in {\mathbf{H}}_{1} \), then\n\n\[ \begin{Vmatrix}{\left( {{A}_{n} \otimes I}\ri... | Proof. See Exercise 9. ∎ | No |
Theorem 20.1 (Trotter Product Formula) Suppose that \( A \) and \( B \) are self-adjoint operators on \( \mathbf{H} \) and that \( A + B \) is densely defined and essentially self-adjoint on \( \operatorname{Dom}\left( A\right) \cap \operatorname{Dom}\left( B\right) \). Then the following results hold.\n\n1. For all \(... | Proof. Since all the operators in Point 1 of the theorem are unitary, it is easy to see that if the result holds on some dense subspace \( W \) of \( \mathbf{H} \), it holds on all of \( \mathbf{H} \). In Point 2 of the theorem, we first make a simple reduction to the case where \( A \) and \( B \) are non-negative, an... | Yes |
Theorem 20.2 (Wiener) For each vector \( {\mathbf{x}}_{0} \in {\mathbb{R}}^{n} \) and each pair of positive numbers \( \sigma \) and \( t \), there exists a unique measure \( {\mu }_{{\mathbf{x}}_{0}}^{\sigma } \) on the Borel \( \sigma \) - algebra in \( {\mathcal{C}}_{{\mathbf{x}}_{0}}\left( {\left\lbrack {0, t}\righ... | Note that the right-hand side of (20.17) is extremely similar to the righthand side of (20.14), except that there are no terms involving the potential \( V \) in the exponent in (20.17). Thus, it is reasonable to think that the Wiener measure is a rigorous version of the formal expression in (20.16). It should be noted... | No |
Proposition 20.5 Suppose \( V : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is bounded and continuous. Then for all \( \phi ,\psi \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \), we have\n\n\[ \left\langle {\phi ,{e}^{-t\widehat{H}/\hslash }\psi }\right\rangle \]\n\n\[ = {\int }_{\mathcal{C}\left( {\left\lbrack {0, t}\righ... | Proof. We begin with (20.14) and apply Theorem 20.2 with parameters chosen as follows. We take \( \sigma = \hslash /m \), we take the sequence \( \left\langle {t}_{j}\right\rangle \) to be given by \( {t}_{j} = {jt}/N \), and we take \( f \) to be the function given by\n\n\[ f\left( {{\mathbf{x}}_{1},{\mathbf{x}}_{2},\... | Yes |
For any smooth manifold \( M \), define a 1-form \( \theta \) on the cotangent bundle \( {T}^{ * }M \) by\n\n\[ \theta {\left( X\right) }_{\left( x,\phi \right) } = \phi \left( {{\pi }_{ * }\left( X\right) }\right) \]\n\nfor each tangent vector \( X \in {T}_{\left( x,\phi \right) }\left( {{T}^{ * }M}\right) \), where \... | Proof. Using a coordinate system \( \left\{ {x}_{j}\right\} \) on \( X \) and the associated standard coordinate system \( \left\{ {{x}_{j},{p}_{j}}\right\} \) on \( {T}^{ * }M \), the projection \( \pi \) is given by \( \pi \left( {x, p}\right) = x \) . Meanwhile, a tangent vector \( X \) to \( {T}^{ * }M \) is expres... | No |
If \( \omega \) is the canonical 2-form on \( {T}^{ * }M \), then the associated Poisson bracket may be computed in standard coordinates as\n\n\[ \n\{ f, g\} = \frac{\partial f}{\partial {x}_{j}}\frac{\partial g}{\partial {p}_{j}} - \frac{\partial f}{\partial {p}_{j}}\frac{\partial g}{\partial {x}_{j}} \n\] \n\nfor all... | Proof. The linear functional \n\n\[ \n\omega \left( {\frac{\partial }{\partial {x}_{j}}, \cdot }\right) \n\] \n\nhas a value of -1 on the vector \( \partial /\partial {p}_{j} \) and a value of 0 on all the other basic partial derivatives. This means that \( \omega \left( {\partial /\partial {x}_{j}, \cdot }\right) = - ... | Yes |
Proposition 21.5 For any smooth functions \( f, g, h \) on \( N \), we have\n\n\[ \{ g, f\} = - \{ f, g\} \]\n\nand\n\n\[ \{ f,{gh}\} = \{ f, g\} h + g\{ f, h\} . \] | Proof. Since \( \omega \) is skew-symmetric on the tangent space to \( N \) at each point and \( {\omega }^{-1} \) is obtained from \( \omega \) by means of an isomorphism of tangent and cotangent space, \( {\omega }^{-1} \) is a skew-symmetric form on the cotangent space. The skew-symmetry of the Poisson bracket follo... | Yes |
Proposition 21.7 For all \( f \) and \( g \) ,\n\n\[ \n{X}_{f}\left( g\right) = \{ f, g\} = - {X}_{g}\left( f\right) .\n\]\n\nFurthermore,\n\n\[ \n\omega \left( {{X}_{f},{X}_{g}}\right) = - \{ f, g\} .\n\] | Proof. For each \( z \in N \), we are using \( \omega \) to identify \( {T}_{z}N \) with \( {T}_{z}^{ * }N \) . Equation (21.8) says that under this identification, \( {X}_{f} \) is identified with \( {df} \) . Thus,\n\n\[ \n- {\omega }^{-1}\left( {{df},{dg}}\right) = - \omega \left( {{X}_{f},{X}_{g}}\right) = - {df}\l... | Yes |
Proposition 21.9 For any smooth function \( f \) on \( N \), the Hamiltonian flow \( {\Phi }^{f} \) preserves \( \omega \) . | Proof. In general, a flow \( \Phi \) preserves a differential form \( \alpha \) if and only if the Lie derivative \( {L}_{X}\alpha = 0 \), where \( X \) is the vector field generating \( \Phi \) . In our case, since \( \omega \) is closed, we have, by (21.7), \[ {\mathcal{L}}_{{X}_{f}}\omega = d\left\lbrack {{i}_{{X}_{... | Yes |
Proposition 21.10 For any smooth functions \( f, g, h \) on \( N \), the Jacobi identity holds:\n\n\[ \{ f,\{ g, h\} \} + \{ g,\{ h, f\} \} + \{ h,\{ f, g\} \} = 0. \] | Proof. Since the Hamiltonian flow \( {\Phi }^{f} \) preserves \( \omega \), it also preserves \( {\omega }^{-1} \) and thus\n\n\[ {\omega }^{-1}\left( {d\left( {g \circ {\Phi }_{t}^{f}}\right), d\left( {h \circ {\Phi }_{t}^{f}}\right) }\right) = {\omega }^{-1}\left( {{dg},{dh}}\right) \circ {\Phi }_{t}^{f}, \]\n\nor, e... | Yes |
Proposition 21.11 For any smooth functions \( f \) and \( g \) on \( N \), the Hamiltonian vector fields \( {X}_{f} \) and \( {X}_{g} \) satisfy\n\n\[ \left\lbrack {{X}_{f},{X}_{g}}\right\rbrack = {X}_{\{ f, g\} } \] | Proof. See Exercise 3. ∎ | No |
Proposition 21.12 Suppose \( \Phi \) is the flow generated by a vector field \( - X \) on \( N \) . If \( \Phi \) preserves \( \omega \) then \( X \) can be represented locally in the form \( X = \) \( {X}_{f} \) for some smooth function \( f \) on \( N \) . If \( N \) is simply connected, the function \( f \) exists g... | Proof. The statement that \( \Phi \) preserves \( \omega \) can be expressed infinitesimally as\n\n\[ \n{\mathcal{L}}_{X}\omega = 0.\n\]\n\nSince also \( \omega \) is closed,(21.7) tells us that\n\n\[ \nd\left( {{i}_{X}\omega }\right) = 0.\n\]\n\nSince \( {i}_{X}\omega \) is closed, this 1 -form can be expressed locall... | Yes |
Proposition 21.14 For any Hamiltonian system \( \left( {N,{\Phi }^{H}}\right) \), we have\n\n\[ \frac{d}{dt}f\left( {{\Phi }_{t}^{H}\left( z\right) }\right) = \{ f, H\} \left( {{\Phi }_{t}^{H}\left( z\right) }\right) ,\]\n\nfor all \( z \in N \), or, more concisely,\n\n\[ \frac{df}{dt} = \{ f, H\} \]\n\nIn particular, ... | Proof. For the flow generated by any vector field \( X \), we have\n\n\[ \frac{d}{dt}f\left( {{\Phi }_{t}\left( z\right) }\right) = {X}_{{\Phi }_{t}\left( z\right) }f \]\n\nIf \( X = - {X}_{f} \), then by Proposition 21.7, we have the claimed result. | No |
Proposition 21.15 A smooth function \( f \) is a conserved quantity for a Hamiltonian system \( \left( {N,{\Phi }^{H}}\right) \) if and only if \( H \) is invariant under the Hamiltonian flow \( {\Phi }^{f} \) generated by \( f \) . | Proof. By the previous proposition, \( H \) is invariant under the flow generated by \( f \) if and only if \( \{ H, f\} = 0 \), which holds if and only if \( \{ f, H\} = 0 \), which holds if and only if \( f \) is a conserved quantity. 1 | Yes |
Theorem 21.17 (Liouville’s Theorem) For any smooth function \( f \) on \( N \), the Hamiltonian flow \( {\Phi }^{f} \) preserves \( \lambda \) . | Proof. The flow \( {\Phi }^{f} \) will preserve \( \lambda \) if and only if the vector field \( {X}_{f} \) satisfies \( {\mathcal{L}}_{{X}_{f}}\lambda = 0 \) . But\n\n\[ \n{\mathcal{L}}_{{X}_{f}}\lambda = \frac{1}{n!}\left\lbrack {\left( {{\mathcal{L}}_{{X}_{f}}\omega }\right) \land \omega \land \cdots \land \omega }\... | Yes |
Proposition 22.3 For any symplectic potential \( \theta \), let \( {\nabla }_{X} \) denote the associated covariant derivative in (22.4). Then for all smooth vector fields \( X \) and \( Y \) on \( {\mathbb{R}}^{2n} \), we have\n\n\[ \left\lbrack {{\nabla }_{X},{\nabla }_{Y}}\right\rbrack = {\nabla }_{\left\lbrack X, Y... | Proof. Using the easily verified identity \( \left\lbrack {{\nabla }_{X}, f}\right\rbrack = X\left( f\right) \), we obtain\n\n\[ \left\lbrack {{\nabla }_{X},{\nabla }_{Y}}\right\rbrack - {\nabla }_{\left\lbrack X, Y\right\rbrack } = - \frac{i}{\hslash }\left\lbrack {X\left( {\theta \left( Y\right) }\right) - Y\left( {\... | Yes |
If \( \theta = {p}_{j}d{x}_{j} \), the prequantized position and momentum operators are given by\n\n\[ \n{Q}_{\text{pre }}\left( {x}_{j}\right) = {x}_{j} + i\hslash \frac{\partial }{\partial {p}_{j}} \]\n\n\[ \n{Q}_{\text{pre }}\left( {p}_{j}\right) = - i\hslash \frac{\partial }{\partial {x}_{j}}. \]\n\nThese operators... | Proof. We compute that \( {X}_{{x}_{j}} = \partial /\partial {p}_{j} \) and that \( \theta \left( {X}_{{x}_{j}}\right) = 0 \), giving the indicated expression for \( {Q}_{\text{pre }}\left( {x}_{j}\right) \) . Meanwhile, \( {X}_{{p}_{j}} = - \partial /\partial {x}_{j} \) and \( \theta \left( {X}_{{p}_{j}}\right) = \) \... | Yes |
Proposition 22.5 Suppose that \( {\theta }_{1} \) and \( {\theta }_{2} \) are two different symplectic potentials for the canonical 2 -form \( \omega \), so that \( d\left( {{\theta }^{1} - {\theta }^{2}}\right) = 0 \) . Let the associated covariant derivatives be denoted by \( {\nabla }^{1} \) and \( {\nabla }^{2} \) ... | Proof. The operation of multiplication by \( {\theta }^{1}\left( X\right) \) commutes with multiplication by \( {e}^{-{i\gamma }/\hslash } \), whereas\n\n\[ \nX\left( {{e}^{{i\gamma }/\hslash }\psi }\right) = {e}^{{i\gamma }/\hslash }{X\psi } + \frac{i}{\hslash }{e}^{{i\gamma }/\hslash }X\left( \gamma \right) \psi .\n\... | Yes |
Proposition 22.6 Consider a harmonic oscillator Hamiltonian of the form\n\n\[ \nH\left( {x, p}\right) = \frac{1}{2m}\left( {{p}^{2} + {\left( m\omega x\right) }^{2}}\right) .\n\]\n\nThen for each integer \( n \), the number \( n\hslash \omega \) is an eigenvalue for \( {Q}_{\mathrm{{pre}}}\left( H\right) \) . | Proof. We can write \( H \) as\n\n\[ \nH\left( {x, p}\right) = \frac{1}{2m}\left( {{p}^{2} + {y}^{2}}\right)\n\]\n\nwhere \( y = {m\omega x} \) . The flow associated to this Hamiltonian consists of rotations in the \( \left( {y, p}\right) \) -plane. If we choose our symplectic potential to be\n\n\[ \n\theta = \frac{1}{... | Yes |
Proposition 22.8 Take the symplectic potential \( \theta = {p}_{j}d{x}_{j} \) . Then the position, momentum, and holomorphic subspaces may be computed as follows. The position subspace consists of smooth functions \( \psi \) on \( {\mathbb{R}}^{2n} \) of the form\n\n\[ \psi \left( {\mathbf{x},\mathbf{p}}\right) = \phi ... | Proof. Since \( \theta \left( {\partial /\partial {p}_{j}}\right) = 0 \), we have \( {\nabla }_{\partial /\partial {p}_{j}} = \partial /\partial {p}_{j} \), so that functions that are covariantly constant in the \( \mathbf{p} \) -directions are actually constant in the \( \mathbf{p} \) -directions. Meanwhile, \( \theta... | Yes |
Proposition 22.11 The position, momentum, and holomorphic subspaces in Definition 22.7 are all invariant under the prequantum operators \( {Q}_{\mathrm{{pre}}}\left( {x}_{j}\right) \) and \( {Q}_{\mathrm{{pre}}}\left( {p}_{j}\right) \) . Specifically, in the position subspace, we have\n\n\[ \n{Q}_{\text{pre }}\left( {x... | Proof. See Exercise 4. ∎ | No |
The position subspace is invariant under \( {Q}_{\mathrm{{pre}}}\left( f\right) \) whenever \( f \) is of the form\n\n\[ f\left( {\mathbf{x},\mathbf{p}}\right) = a\left( \mathbf{x}\right) + {b}_{j}\left( \mathbf{x}\right) {p}_{j} \]\n\nfor some smooth functions \( a \) and \( {b}_{1},\ldots ,{b}_{n} \) on \( {\mathbb{R... | Proof. If \( f \) is of the form (22.14), calculation shows that \( \theta \left( {X}_{f}\right) + f = a\left( \mathbf{x}\right) \) . If we drop any terms in \( {X}_{f} \) involving \( \partial /\partial {p}_{j} \), since these are zero on the position subspace, we end up with\n\n\[ {Q}_{\text{pre }}\left( f\right) \le... | Yes |
Proposition 22.13 For any \( \alpha > 0 \), let \( {\mathbf{H}}_{\alpha } \) be the subspace of \( {L}^{2}\left( {\mathbb{R}}^{2n}\right) \) consisting of smooth functions \( \psi \) that satisfy \( {\nabla }_{\partial /\partial {\bar{z}}_{j}}\psi = 0 \), where \( \partial /\partial {\bar{z}}_{j} \) is as in (22.9). Th... | Proof. The invariance of \( {\mathbf{H}}_{\alpha } \) is a simple calculation (Exercise 5). Irreducibility can be established by reducing to the previously established irreducibility of the Segal-Bargmann space under the operators \( {T}_{\mathbf{a}} \) in Theorem 14.16. To this end, we should check that the unitary ma... | No |
Consider a harmonic oscillator with Hamiltonian\n\n\[ H = \frac{1}{2m}\left( {{p}^{2} + {\left( m\omega x\right) }^{2}}\right) .\n\]\n\nConsider also the subspace \( {\mathbf{H}}_{\alpha } \) in Proposition 22.13, with \( \alpha = 1/\left( {m\omega }\right) \). Then the operator \( {Q}_{\mathrm{{pre}}}\left( H\right) \... | Proof. As in the proof of Proposition 22.6, we introduce the variable \( y = {m\omega x} \). With \( \alpha = 1/\left( {m\omega }\right) \), this gives \( z = \left( {y - {ip}}\right) /\left( {m\omega }\right) \). We use the symplectic potential\n\n\[ \theta = \frac{1}{2}\left( {{pdx} - {xdp}}\right) = \frac{1}{2m\omeg... | Yes |
Proposition 23.5 Let \( {s}_{0} \) be a local isometric trivialization of \( L \) and let \( \theta \) be the associated connection 1-form. Then the curvature 2-form \( \omega \) of \( \nabla \) is expressed locally as\n\n\[ \omega = {d\theta }\text{.} \] \n\nIn particular, \( \omega \) is a closed 2-form. | Proof. The computation is precisely the same as in the proof of Proposition 22.3 in the Euclidean case. | No |
Proposition 23.6 Let \( \\left( {L,\\nabla }\\right) \) be a Hermitian line bundle with connection over \( N \) with curvature 2 -form \( \\omega \) . For each point \( {z}_{0} \\in N \) and 1 -form \( \\theta \) defined in a neighborhood \( U \) of \( {z}_{0} \) satisfying \( {d\\theta } = \\omega \), there is a subne... | Proof. Let \( {s}_{0} \) be any isometric trivializing section defined in a neighborhood of \( {z}_{0} \) and let \( \\eta \) be the associated connection 1-form. Since \( d\\left( {\\eta - \\theta }\\right) = 0 \) ,\n\nthere is a subneighborhood \( V \\subset U \) of \( {z}_{0} \) on which \( \\eta - \\theta = {df} \)... | Yes |
Proposition 23.7 If \( \left( {{L}_{1},{\nabla }^{1}}\right) \) and \( \left( {{L}_{2},{\nabla }^{2}}\right) \) are Hermitian line bundles with connection over \( N \), let \( {L}_{1} \otimes {L}_{2} \) denote the line bundle over \( N \) for which the fiber over \( x \) is \( {L}_{1, x} \otimes {L}_{2, x} \), with the... | The proof of this proposition is a straightforward exercise in \ | No |
Theorem 23.9 Suppose \( \omega \) is a closed 2-form on a manifold \( N \) for which \( \omega /\left( {2\pi }\right) \) is integral in the sense of (23.8). Then there exists a Hermitian line bundle \( L \) over \( N \) with Hermitian connection \( \nabla \) such that the curvature of \( \nabla \) is equal to \( \omega... | See Sect. 8.3 of [45] for a proof of this result. | No |
Proposition 23.13 If \( f \) is real-valued, then \( {Q}_{\text{pre }}\left( f\right) \) is symmetric on the space of smooth compactly supported sections of \( L \) . | Proof. Let \( {s}_{1} \) and \( {s}_{2} \) be smooth, compactly supported sections of \( L \) and let \( {\Phi }^{f} \) denote the Hamiltonian flow generated by \( f \) . For all sufficiently small \( t \), every point in the supports of \( {s}_{1} \) and \( {s}_{2} \) will contained in the domain of \( {\Phi }_{t}^{f}... | Yes |
Proposition 23.14 For any \( f, g \in {C}^{\infty }\left( X\right) \), we have\n\n\[ \frac{1}{i\hslash }\left\lbrack {{Q}_{\text{pre }}\left( f\right) ,{Q}_{\text{pre }}\left( g\right) }\right\rbrack = {Q}_{\text{pre }}\left( {\{ f, g\} }\right) ,\]\n\nwhere the equality holds as operators on the space of smooth sectio... | Proof. The argument is precisely the same as in Proposition 22.1 in the \( {\mathbb{R}}^{2n} \) case. ∎ | No |
If \( M \) is any smooth manifold, let \( N = {T}^{ * }M \) be the cotangent bundle of \( M \), equipped with the canonical 2-form \( \omega \) (Example 21.2). For each \( z \in {T}^{ * }M \), let \( {P}_{z} \) be the complexification of the tangent space to the fiber \( {T}_{z}^{ * }M \) . Then \( P \) is a polarizati... | Proof. If \( \left\{ {x}_{j}\right\} \) is any local coordinate system on \( M \), let \( \left\{ {{x}_{j},{p}_{j}}\right\} \) be the associated local coordinate system on \( {T}^{ * }M \) . The canonical 2 -form is given by \( \omega = d{p}_{j} \land d{x}_{j} \) . At each point \( z \in {T}^{ * }M \), the vertical sub... | Yes |
Proposition 23.18 Suppose \( P \) is a purely complex polarization on \( N \) . For each \( z \in N \), let \( {J}_{z} : {T}_{z}^{\mathbb{C}}N \rightarrow {T}_{z}^{\mathbb{C}}N \) be the unique linear map such that \( {J}_{z} = \) \( {iI} \) on \( {P}_{z} \) and \( {J}_{z} = - {iI} \) on \( \overline{{P}_{z}} \) . Then... | Proof. Since the restriction of \( {J}_{z} \) to \( \overline{{P}_{z}} \) is the complex-conjugate of its restriction to \( {P}_{z} \), the map \( {J}_{z} \) commutes with complex conjugation and thus maps real vectors (those satisfying \( \bar{X} = X \) ) to real vectors. Meanwhile, since \( {P}_{z} \) is Lagrangian a... | Yes |
If \( N = {T}^{ * }M \) for some manifold \( M \) and \( P \) is the vertical polarization on \( N \), then a Hamiltonian vector field \( {X}_{f} \) preserves \( P \) if and only if \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} \) is constant on each fiber and \( {f}_{2} \) is linear on each fiber. | In local coordinates \( \left\{ {{x}_{j},{p}_{j}}\right\} \), a vector field \( X \) lying in \( P \) has the form \( X = {g}_{j}\partial /\partial {p}_{j} \) . Thus,\n\n\[ \left\lbrack {{X}_{f}, X}\right\rbrack = \left\lbrack {\frac{\partial f}{\partial {p}_{j}}\frac{\partial }{\partial {x}_{j}},{g}_{k}\frac{\partial ... | No |
Theorem 23.24 For any smooth, complex-valued function \( f \) on \( N \), if the Hamiltonian vector field \( {X}_{f} \) preserves \( \bar{P} \), then \( f \) is quantizable. | Proof. Given a polarized section \( s \), we apply \( {Q}_{\text{pre }}\left( f\right) \) to \( s \) and then test whether \( {Q}_{\text{pre }}\left( f\right) s \) is still polarized, by applying \( {\nabla }_{X} \) for some vector field \( X \) lying in \( \bar{P} \) . To this end, it is useful to compute the commutat... | Yes |
Proposition 23.25 If \( f \) is a smooth, complex-valued function on \( N \) and the derivatives of \( f \) in the \( \bar{P} \) directions are zero, then \( {Q}_{\mathrm{{pre}}}\left( f\right) \) preserves the space \( P \) -polarized sections, and the restriction of \( {Q}_{\mathrm{{pre}}}\left( f\right) \) to this s... | Proof. If the derivatives of \( f \) in the direction of \( \bar{P} \) are zero, then for \( X \in \bar{P} \) , we have\n\n\[ 0 = X\left( f\right) = {df}\left( X\right) = \omega \left( {{X}_{f}, X}\right) \]\n\nmeaning that \( {X}_{f} \) is in the \( \omega \) -orthogonal complement of \( \bar{P} \) . But since \( \bar... | Yes |
Proposition 23.26 If \( P \) is a purely real polarization on \( N \), then for any \( {z}_{0} \in N \), there exist a neighborhood \( U \) of \( {z}_{0} \) and a \( P \) -polarized section \( s \) of \( L \) defined over \( U \) such that \( s\left( {z}_{0}\right) \neq 0 \) . | Proof. According to the local form of the Frobenius theorem, we can find a neighborhood \( U \) of \( {z}_{0} \) and a diffeomorphism \( \Phi \) of \( U \) with a neighborhood \( V \) of the origin in \( {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) such that under \( \Phi \), the polarization \( P \) looks like the vert... | Yes |
Let \( N = {S}^{1} \times \mathbb{R} \), equipped with the symplectic form \( \omega = \) \( {dx} \land {d\phi } \), where \( x \) is the linear coordinate on \( \mathbb{R} \) and \( \phi \) is the angular coordinate on \( {S}^{1} \) . Let \( L \) be the trivial line bundle on \( N \), with sections that are identified... | Proof. If we define a section locally on a given leaf \( {S}^{1} \times \{ x\} \) as\n\n\[ s\left( \phi \right) = c{e}^{{ix\phi }/\hslash } \]\n\nfor some nonzero constant \( c \), then it is easily verified that \( {\nabla }_{\partial /\partial \phi }s = 0 \) . After one trip around the circle, the value of this secti... | Yes |
Example 23.30 Let \( N \) be the unit disk \( D \subset {\mathbb{R}}^{2} \) equipped with the following symplectic form:\n\n\[ \omega = 4{\left( 1 - {\left| z\right| }^{2}\right) }^{-2}{dx} \land {dy} = {\left( 1 - {r}^{2}\right) }^{-2}{rdr} \land {d\phi }, \]\n\nwhere \( \left( {r,\phi }\right) \) are the usual polar ... | Proof. See Exercise 8. ∎ | No |
Theorem 23.33 Assume \( N \) is compact and let \( P \) be a Kähler polarization on \( N \) . For each positive integer \( k \), let \( {\mathbf{H}}_{k} \) denote the space of polarized sections of \( {L}^{\otimes k} \) . Then for all \( k,{\mathbf{H}}_{k} \) is finite dimensional. Furthermore, for all sufficiently lar... | The finite dimensionality of \( {\mathbf{H}}_{k} \) is a standard result in the theory of compact, complex manifolds. The embedding of \( N \) into \( \mathcal{P}\left( {\mathbf{H}}_{k}\right) \) is the Kodaira embedding theorem, which we will not prove here. | No |
Lemma 23.34 Let \( N \) be a complex manifold with almost-complex structure \( J \) and let \( \omega \) be a closed, \( J \) -invariant, real-valued \( \left( {1,1}\right) \) -form on \( N \) . Then for every point \( {z}_{0} \in N \), there exists a smooth, real-valued function \( \kappa \) defined in a neighborhood ... | Proof. By assumption, \( {d\omega } = \left( {\partial + \bar{\partial }}\right) \omega = 0 \), from which it follows that \( \partial \omega = \bar{\partial }\omega = 0 \), because \( \partial \omega \) is a \( \left( {2,1}\right) \) -form and \( \bar{\partial }\omega \) is a \( \left( {1,2}\right) \) form. Thus, by t... | Yes |
Let \( N = {T}^{ * }{\mathbb{R}}^{n} \cong {\mathbb{R}}^{2n} \) and let \( P \) be the vertical polarization on \( N \) . Then an \( n \) -form \( \alpha \) on \( {\mathbb{R}}^{2n} \) is a section of \( {\mathcal{K}}_{P} \) if and only if \( \alpha \) is of the form\n\n\[ \alpha = f\left( {\mathbf{x},\mathbf{p}}\right)... | Proof. If \( \alpha \) contained any term involving \( d{p}_{j} \), the contraction of \( \alpha \) with \( \partial /\partial {p}_{j} \) would not be zero, leaving (23.18) as the only possible form for a section of \( {\mathcal{K}}_{P} \) . Assuming \( \alpha \) is of the form (23.18), if \( f \) is not independent of... | Yes |
Proposition 23.37 If the leaf space \( \Xi \) of \( P \) is a smooth manifold and \( \alpha \) is a polarized section of \( {\mathcal{K}}_{P} \), then there exists a unique \( n \) -form \( \widetilde{\alpha } \) on \( \Xi \) such that\n\n\[ \alpha = {q}^{ * }\left( \widetilde{\alpha }\right) \]\n\nwhere \( q : N \righ... | Proof. Suppose, first, that \( \alpha = {q}^{ * }\left( \beta \right) \), for an \( n \) -form \( \beta \) on \( \Xi \) . Then \( X\lrcorner \alpha = 0 \) whenever \( X \) lies in \( P \), since \( P \) is the kernel of \( {q}_{ * } \) . Furthermore, \( {d\alpha } = \) \( {q}^{ * }\left( {d\beta }\right) = 0 \), since ... | No |
Proposition 23.38 Suppose \( X \) is a vector field on \( N \) that preserves \( P \) , in the sense of Definition 23.22, and suppose \( \alpha \) is a smooth section of \( {\mathcal{K}}_{P} \) . Then the Lie derivative \( {\mathcal{L}}_{X}\alpha \) is another section of \( {\mathcal{K}}_{P} \) and if \( \alpha \) is p... | Proof. Suppose \( {X}_{1},\ldots ,{X}_{n} \) are smooth vector fields, with \( {X}_{1} \) lying in \( \bar{P} = P \) . Then, by a standard formula for the Lie derivative,\n\n\[ \left( {{\mathcal{L}}_{X}\alpha }\right) \left( {{X}_{1},\ldots ,{X}_{n}}\right) \]\n\n\[ = X\left( {\alpha \left( {{X}_{1},\ldots ,{X}_{n}}\ri... | Yes |
Proposition 23.39 Suppose the leaf space \( \\Xi \) of \( P \) is a smooth manifold and that a vector field \( X \) on \( N \) preserves \( P \) . Then there exists a unique vector field \( Y \) on \( \\Xi \) such that\n\n\[ \n{q}_{*, z}\\left( X\\right) = Y \n\]\n\n(23.21)\n\nfor all \( z \\in N \) . Furthermore, if \... | Proof. By Definition 23.22, \( \\left\\lbrack {X, Z}\\right\\rbrack \) lies in \( P \) whenever the vector field \( Z \) lies in \( P \) . Thus, if a function \( \\phi \) is constant along \( P \) (i.e., annihilated by every vector field \( Z \) lying in \( P \) ), the same will be true of \( {X\\phi } \) . Thus, if \(... | Yes |
Proposition 23.41 Let \( {\delta }_{P} \) be a fixed square root of \( {\mathcal{K}}_{P} \). For any vector field \( X \) lying in \( P \), there is a unique linear operator \( {\nabla }_{X} \) mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \), such that\n\n\[ \n{\nabla }_{X}\left( {f{s}_{1}}\r... | Proof. If \( V \) is a one-dimensional vector space, then the map \( \otimes : V \times V \rightarrow \) \( V \otimes V \) is commutative: \( u \otimes v = v \otimes u \) for all \( u, v \in V \). Furthermore, if \( {u}_{0} \) is a nonzero element of \( V \), then the map \( u \mapsto u \otimes {u}_{0} \) is an inverti... | Yes |
Example 23.45 Let the notation be as in Example 23.43, and let \( f : {\mathbb{R}}^{2} \rightarrow \) \( \mathbb{R} \) be of the form\n\n\[ f\left( {x, p}\right) = a\left( x\right) + b\left( x\right) p \]\n\nfor some smooth functions \( a \) and \( b \) on \( \mathbb{R} \) . Then \( {X}_{f} \) preserves \( P \) and\n\n... | Proof. We have computed \( {Q}_{\text{pre }}\left( f\right) \) in (22.15) in the proof of Proposition 22.12. We compute that \( {\widetilde{X}}_{f} \) is equal to \( - b\left( x\right) \partial /\partial x \) plus a term involving \( \partial /\partial p \) . Since the 1 -form \( {dx} \) is closed, we obtain, by (21.7)... | Yes |
Theorem 23.46 Suppose \( f \) and \( g \) are functions on \( N \) for which \( {X}_{f} \) and \( {X}_{g} \) preserve \( P \) . Then the operators \( Q\left( f\right) \) and \( Q\left( g\right) \) satisfy\n\n\[ \frac{1}{i\hslash }\left\lbrack {Q\left( f\right), Q\left( g\right) }\right\rbrack = Q\left( {\{ f, g\} }\rig... | Proof. Since \( Q\left( h\right) \) is a local operator for any function \( h \), it suffices to prove the result locally. Let us choose, then, a local nonvanishing section \( {\nu }_{0} \) of \( {\delta }_{P}^{\mathbb{C}} \), so that, locally, each section \( s \) of \( L \otimes {\delta }_{P}^{\mathbb{C}} \) can be d... | Yes |
Theorem 23.47 If \( f \in {C}^{\infty }\left( N\right) \) is real valued and \( {X}_{f} \) preserves \( P \), then the operator \( Q\left( f\right) \) is symmetric on the space of smooth sections \( s \) in the half-form space for which \( \left( {s, s}\right) \) has compact support on \( \Xi \) . | Proof. Suppose \( \alpha = {q}^{ * }\left( \beta \right) \) is polarized section of \( {\mathcal{K}}_{P} \), so that there is, at least locally, a corresponding polarized section \( \sqrt{{q}^{ * }\left( \beta \right) } \) of \( {\delta }_{P} \) . If \( {X}_{f} \) preserves \( P \), then by Proposition 23.39, there is ... | No |
Suppose now that \( f \) is a function on \( {T}^{ * }M \) of the form \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} \) is constant on each fiber of \( {T}^{ * }M \) and \( {f}_{2} \) is linear on each fiber. Then \( {f}_{2} \) may be thought of as a section of \( {T}^{* * }M \cong {TM} \), that is, as a vector field \... | Proof. The calculation is precisely the same as in the proof of Theorem 23.47 , except that the decomposition in (23.37) is now global. The claimed form of \( Q\left( f\right) \) is nothing but the expression (23.38), where the reader may easily compute, using local coordinates, that \( - \theta \left( {X}_{f}\right) -... | No |
Proposition 23.50 If \( \alpha \) is an \( \left( {n,0}\right) \) -form on \( N \), then at each point the \( {2n} \) -form\n\n\[ \n{\left( -1\right) }^{n\left( {n - 1}\right) /2}{\left( -i\right) }^{n}\bar{\alpha } \land \alpha \n\]\n\nis a non-negative multiple of the Liouville form \( \lambda \) . There is then a un... | Proof. See Exercise 17. - | No |
Consider \( {\mathbb{R}}^{2} \cong {T}^{ * }\mathbb{R} \) with the Kähler polarization \( P \) given by the global complex coordinate \( z = \left( {x - {ip}/\left( {m\omega }\right) }\right) \), for some positive number \( \omega \) . Take \( {\delta }_{P} \) to be trivial with trivializing section \( \sqrt{dz} \) . C... | Proof. The calculation is the same as in the proof of Proposition 22.14, except for the addition of the Lie derivative term. A simple calculation shows that \( {\mathcal{L}}_{{X}_{H}}\left( {dz}\right) = {i\omega dz} \), from which it follows that \( {\mathcal{L}}_{{X}_{H}}\sqrt{dz} = \) \( \left( {{i\omega }/2}\right)... | Yes |
Example 23.54 Consider \( N = {\mathbb{R}}^{2} \cong {T}^{ * }\mathbb{R} \) and take \( L \) to be trivial with connection 1-form \( \theta = {pdx} \) . Let \( P \) be the vertical polarization, spanned at each point by \( \partial /\partial p \), and let \( {P}^{\prime } \) be the horizontal polarization, spanned at e... | Proof. The forms (23.44) and (23.45) are obtained by a simple modification of the argument in the proof of Proposition 22.8. We can compute that the pointwise pairing of \( \sqrt{dx} \) and \( \sqrt{dp} \) is -1, which gives the indicated form of the pairing in (23.46). The pairing may be rewritten as\n\n\[ \n{\int }_{... | Yes |
Lemma 1.1.1. Let \( K \subseteq \Omega \subseteq {\mathbb{C}}^{n} \) be compact. There is a constant \( {C}_{K} > 0 \) , depending on \( K \) and on \( n \), such that\n\n\[ \mathop{\sup }\limits_{{z \in K}}\left| {f\left( z\right) }\right| \leq {C}_{K}\parallel f{\parallel }_{{A}^{2}\left( \Omega \right) },\text{ all ... | Proof. Since \( K \) is compact, there is an \( r\left( K\right) = r > 0 \) so that, for any \( z \in \) \( K, B\left( {z, r}\right) \subseteq \Omega \) . Here \( B\left( {z, r}\right) \) is the usual Euclidean ball with center \( z \) and radius \( r \) .\n\nTherefore for each \( z \in K \) and \( f \in {A}^{2}\left( ... | Yes |
Lemma 1.1.2. The space \( {A}^{2}\left( \Omega \right) \) is a Hilbert space with the inner product \( \langle f, g\rangle \equiv \) \( {\int }_{\Omega }f\left( z\right) \overline{g\left( z\right) }\mathrm{d}V\left( z\right) . \) | Proof. Everything is clear except for completeness. Let \( \left\{ {f}_{j}\right\} \subseteq {A}^{2} \) be a sequence that is Cauchy in norm. Since \( {L}^{2} \) is complete there is an \( {L}^{2} \) limit function \( f \) . We need to see that \( f \) is holomorphic. But Lemma 1.1.1 yields that norm convergence implie... | No |
Lemma 1.1.3. For each fixed \( z \in \Omega \), the functional\n\n\[ \n{\Phi }_{z} : f \mapsto f\left( z\right) ,\;f \in {A}^{2}\left( \Omega \right) \n\]\n\nis a continuous linear functional on \( {A}^{2}\left( \Omega \right) \) . | Proof. This is immediate from Lemma 1.1.1 if we take \( K \) to be the singleton \( \{ z\} \) . \( ▱ \) | Yes |
Proposition 1.1.5. The Bergman kernel \( K\left( {z,\zeta }\right) \) is conjugate symmetric: \( K\left( {z,\zeta }\right) = \) \( \overline{K\left( {\zeta, z}\right) } \) . | Proof. By its very definition, \( \overline{K\left( {\zeta , \cdot }\right) } \in {A}^{2}\left( \Omega \right) \) for each fixed \( \zeta \) . Therefore the reproducing property of the Bergman kernel gives\n\n\[ \n{\int }_{\Omega }K\left( {z, t}\right) \overline{K\left( {\zeta, t}\right) }\mathrm{d}V\left( t\right) = \... | Yes |
Proposition 1.1.6. The Bergman kernel is uniquely determined by the properties that it is an element of \( {A}^{2}\left( \Omega \right) \) in \( z \), is conjugate symmetric, and reproduces \( {A}^{2}\left( \Omega \right) \) . | Proof. Let \( {K}^{\prime }\left( {z,\zeta }\right) \) be another such kernel. Then\n\n\[ K\left( {z,\zeta }\right) = \overline{K\left( {\zeta, z}\right) } = \int {K}^{\prime }\left( {z, t}\right) \overline{K\left( {\zeta, t}\right) }\mathrm{d}V\left( t\right) \]\n\n\[ = \overline{\int }K\left( {\zeta, t}\right) \overl... | Yes |
Proposition 1.1.7. Let \( L \) be a compact subset of \( \Omega \) . Then the series\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{\infty }{\phi }_{j}\left( z\right) \overline{{\phi }_{j}\left( \zeta \right) } \]\n\nsums uniformly on \( L \times L \) to the Bergman kernel \( K\left( {z,\zeta }\right) \) . | Proof. By the Riesz-Fischer and Riesz representation theorems, we obtain\n\n\[ \mathop{\sup }\limits_{{z \in L}}{\left( \mathop{\sum }\limits_{{j = 1}}^{\infty }{\left| {\phi }_{j}\left( z\right) \right| }^{2}\right) }^{1/2} = \mathop{\sup }\limits_{{z \in L}}{\begin{Vmatrix}{\left\{ {\phi }_{j}\left( z\right) \right\}... | Yes |
Proposition 1.1.9. If \( \Omega \) is a bounded domain in \( {\mathbb{C}}^{n} \), then the mapping\n\n\[ P : f \mapsto {\int }_{\Omega }K\left( {\cdot ,\zeta }\right) f\left( \zeta \right) \mathrm{d}V\left( \zeta \right) \]\n\nis the Hilbert space orthogonal projection of \( {L}^{2}\left( {\Omega ,\mathrm{d}V}\right) \... | Proof. Notice that \( P \) is idempotent and self-adjoint and that \( {A}^{2}\left( \Omega \right) \) is precisely the set of elements of \( {L}^{2} \) that are fixed by \( P \) . | No |
Proposition 1.1.11. With notation as in the definition, we have\n\n\\[ \n\\det {J}_{\\mathbb{R}}f = {\\left| \\det {J}_{\\mathbb{C}}f\\right| }^{2} \n\\]\n\nwhenever \\( f \\) is a holomorphic mapping. | Proof. We exploit the functoriality of the Jacobian. Let \\( w = \\left( {{w}_{1},\\ldots ,{w}_{n}}\\right) = \\) \\( f\\left( z\\right) = \\left( {{f}_{1}\\left( z\\right) ,\\ldots ,{f}_{n}\\left( z\\right) }\\right) \\) . Write \\( {z}_{j} = {x}_{j} + i{y}_{j},{w}_{j} = {\\xi }_{j} + i{\\eta }_{j}, j = 1,\\ldots, n \... | Yes |
Theorem 1.1.12. Let \( {f}_{j}\left( {w, z}\right), j = 1,\ldots, m \) be holomorphic functions of \( \left( {w, z}\right) = \) \( \left( {\left( {{w}_{1},\ldots ,{w}_{m}}\right) ,\left( {{z}_{1},\ldots ,{z}_{n}}\right) }\right) \) near a point \( \left( {{w}^{0},{z}^{0}}\right) \in {\mathbb{C}}^{m} \times {\mathbb{C}}... | Proof. We rewrite the system of equations as\n\n\[ \n\operatorname{Re}{f}_{j}\left( {w, z}\right) = 0,\operatorname{Im}{f}_{j}\left( {w, z}\right) = 0 \n\] \n\nfor the \( {2m} \) real variables \( \operatorname{Re}{w}_{k},\operatorname{Im}{w}_{k}, k = 1,\ldots, m \) . By Proposition 1.1.11, the determinant of the Jacob... | Yes |
Proposition 1.1.14. Let \( {\Omega }_{1},{\Omega }_{2} \) be domains in \( {\mathbb{C}}^{n} \) . Let \( f : {\Omega }_{1} \rightarrow {\Omega }_{2} \) be biholomorphic. Then\n\n\[ \det {J}_{\mathbb{C}}f\left( z\right) {K}_{{\Omega }_{2}}\left( {f\left( z\right), f\left( \zeta \right) }\right) \det \overline{{J}_{\mathb... | Proof. Let \( \phi \in {A}^{2}\left( {\Omega }_{1}\right) \) . Then, by change of variable,\n\n\[ {\int }_{{\Omega }_{1}}\det {J}_{\mathbb{C}}f\left( z\right) {K}_{{\Omega }_{2}}\left( {f\left( z\right), f\left( \zeta \right) }\right) \det \overline{{J}_{\mathbb{C}}f\left( \zeta \right) }\phi \left( \zeta \right) \math... | Yes |
Proposition 1.1.15. For \( z \in \Omega \subset \subset {\mathbb{C}}^{n} \) it holds that \( {K}_{\Omega }\left( {z, z}\right) > 0 \) . | Proof. Now\n\n\[ \n{K}_{\Omega }\left( {z, z}\right) = \mathop{\sum }\limits_{{j = 1}}^{\infty }{\left| {\phi }_{j}\left( z\right) \right| }^{2} \geq 0. \]\n\nIf in fact \( K\left( {z, z}\right) = 0 \) for some \( z \), then \( {\phi }_{j}\left( z\right) = 0 \) for all \( j \) ; hence, \( f\left( z\right) = 0 \) for ev... | Yes |
Proposition 1.1.18. Let \( {\Omega }_{1},{\Omega }_{2} \subseteq {\mathbb{C}}^{n} \) be domains and let \( f : {\Omega }_{1} \rightarrow {\Omega }_{2} \) be a biholomorphic mapping. Then \( f \) induces an isometry of Bergman metrics:\n\n\[ \n{\left| \xi \right| }_{B, z} = {\left| \left( {J}_{\mathbb{C}}f\right) \xi \r... | Proof. This is a formal exercise but we include it for completeness: From the definitions, it suffices to check that\n\n\[ \n\sum {g}_{i, j}^{{\Omega }_{2}}\left( {f\left( z\right) }\right) {\left( {J}_{\mathbb{C}}f\left( z\right) w\right) }_{i}{\left( \overline{{J}_{\mathbb{C}}f\left( z\right) w}\right) }_{j} = \matho... | Yes |
Proposition 1.1.19. Let \( \Omega \subset \subset {\mathbb{C}}^{n} \) be a domain. Let \( z \in \Omega \) . Then\n\n\[ K\left( {z, z}\right) = \mathop{\sup }\limits_{{f \in {A}^{2}\left( \Omega \right) }}\frac{{\left| f\left( z\right) \right| }^{2}}{\parallel f{\parallel }_{{A}^{2}}^{2}} = \mathop{\sup }\limits_{{\para... | Proof. Now\n\n\[ K\left( {z, z}\right) = \sum {\left| {\phi }_{j}\left( z\right) \right| }^{2} \]\n\n\[ = {\left( \mathop{\sup }\limits_{{{\begin{Vmatrix}\left\{ {a}_{j}\right\} \end{Vmatrix}}_{{\ell }^{2}} = 1}}\left| \sum {\phi }_{j}\left( z\right) {a}_{j}\right| \right) }^{2} \]\n\n\[ = \mathop{\sup }\limits_{{\para... | Yes |
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