Non-Relativistic Quantum Mechanics as a Gauge Theory

Transcription

1 Non-Relativistic Quantum Mechanics as a Gauge Theory Sungwook Lee Department of Mathematics, University of Southern Mississippi LA/MS Section of MAA Meeting, March 1, 2013

2 Outline Lifted Quantum Mechanics 1 Lifted Quantum Mechanics 2

3 State Functions Lifted Quantum Mechanics In quantum mechanics, a particle is described by a complex-valued wave function, called state function ψ : R 3+1 C n. The states ψ of a quantum mechanical system forms an innite dimensional Hilbert space. The probability that a particle in a state ψ to be found inside the volume V R 3 is given by ψ ψ = ψ ψd 3 x where ψ = ψ t. While the probability density ψ 2 is an observable, the state function ψ(t, x) itself is not an observable. ψ(r, t) is a manifestation of a particle in a state. So, there is no physical reason why wave functions have to be C n (complex vector)-valued functions! V

4 State Functions Lifted Quantum Mechanics In quantum mechanics, a particle is described by a complex-valued wave function, called state function ψ : R 3+1 C n. The states ψ of a quantum mechanical system forms an innite dimensional Hilbert space. The probability that a particle in a state ψ to be found inside the volume V R 3 is given by ψ ψ = ψ ψd 3 x where ψ = ψ t. While the probability density ψ 2 is an observable, the state function ψ(t, x) itself is not an observable. ψ(r, t) is a manifestation of a particle in a state. So, there is no physical reason why wave functions have to be C n (complex vector)-valued functions! V

5 State Functions Lifted Quantum Mechanics In quantum mechanics, a particle is described by a complex-valued wave function, called state function ψ : R 3+1 C n. The states ψ of a quantum mechanical system forms an innite dimensional Hilbert space. The probability that a particle in a state ψ to be found inside the volume V R 3 is given by ψ ψ = ψ ψd 3 x where ψ = ψ t. While the probability density ψ 2 is an observable, the state function ψ(t, x) itself is not an observable. ψ(r, t) is a manifestation of a particle in a state. So, there is no physical reason why wave functions have to be C n (complex vector)-valued functions! V

6 State Functions Lifted Quantum Mechanics In quantum mechanics, a particle is described by a complex-valued wave function, called state function ψ : R 3+1 C n. The states ψ of a quantum mechanical system forms an innite dimensional Hilbert space. The probability that a particle in a state ψ to be found inside the volume V R 3 is given by ψ ψ = ψ ψd 3 x where ψ = ψ t. While the probability density ψ 2 is an observable, the state function ψ(t, x) itself is not an observable. ψ(r, t) is a manifestation of a particle in a state. So, there is no physical reason why wave functions have to be C n (complex vector)-valued functions! V

7 Holomorphic Tangent Bundle T + (C n ) We regard C n as a Hermitian manifold of complex dimension n with the Hermitian metric g = dz µ d z µ. The complexied tangent bundle of C n, T (C n ) C is decomposed into holomorphic and anti-holomorphic tangent bundles of C T (C n ) C = T + (C n ) T (C n ). The holomorphic tangent bundle T + (C n ) is a holomorphic vector bundle.

8 Holomorphic Tangent Bundle T + (C n ) We regard C n as a Hermitian manifold of complex dimension n with the Hermitian metric g = dz µ d z µ. The complexied tangent bundle of C n, T (C n ) C is decomposed into holomorphic and anti-holomorphic tangent bundles of C T (C n ) C = T + (C n ) T (C n ). The holomorphic tangent bundle T + (C n ) is a holomorphic vector bundle.

9 Holomorphic Tangent Bundle T + (C n ) We regard C n as a Hermitian manifold of complex dimension n with the Hermitian metric g = dz µ d z µ. The complexied tangent bundle of C n, T (C n ) C is decomposed into holomorphic and anti-holomorphic tangent bundles of C T (C n ) C = T + (C n ) T (C n ). The holomorphic tangent bundle T + (C n ) is a holomorphic vector bundle.

10 Lift of a Map Lifted Quantum Mechanics Denition A map h : X Z is called a lift of f : X Y if there exists a map g : Z Y such that f = g h. Z X Y

11 Lifting a State Function Let φ : C n T (C n ) C be a vector eld dened by φ(z µ, z µ ) = z µ z µ + z µ z µ. φ is a section of the complexied tangent bundle T (C n ) C since π φ = Id where π : T (C n ) C C n is the projection map. Given a state function ψ : R 3+1 C n, dene Ψ : R 3+1 T (C n ) C by Ψ = φ ψ. Then Ψ is a lift of ψ since π Ψ = ψ. We call Ψ a lifted state function.

12 Lifting a State Function Let φ : C n T (C n ) C be a vector eld dened by φ(z µ, z µ ) = z µ z µ + z µ z µ. φ is a section of the complexied tangent bundle T (C n ) C since π φ = Id where π : T (C n ) C C n is the projection map. Given a state function ψ : R 3+1 C n, dene Ψ : R 3+1 T (C n ) C by Ψ = φ ψ. Then Ψ is a lift of ψ since π Ψ = ψ. We call Ψ a lifted state function.

13 Lifting a State Function Let φ : C n T (C n ) C be a vector eld dened by φ(z µ, z µ ) = z µ z µ + z µ z µ. φ is a section of the complexied tangent bundle T (C n ) C since π φ = Id where π : T (C n ) C C n is the projection map. Given a state function ψ : R 3+1 C n, dene Ψ : R 3+1 T (C n ) C by Ψ = φ ψ. Then Ψ is a lift of ψ since π Ψ = ψ. We call Ψ a lifted state function.

14 de Broglie Wave Lifted Quantum Mechanics Example Ψ(r, t) = Ae i(k r ωt) z + Āe i(k r ωt) is the lift of the de Broglie z wave ψ(r, t) = Ae i(k r ωt), a plane wave that describes the motion of a free particle with momentum p = k h.

15 The Probability of a Lifted State Recall that ( g z µ, z µ So, we obtain ) ( = g z µ, z µ ψ 2 = g(ψ,ψ). ) ( = 0, g z µ, z µ ) = 1 2. We dene g(ψ,ψ) to be the probability density of the lifeted state function Ψ. Since a state function and its lift have the same probability, we may study quantum mechanics with lifted state functions.

16 The Probability of a Lifted State Recall that ( g z µ, z µ So, we obtain ) ( = g z µ, z µ ψ 2 = g(ψ,ψ). ) ( = 0, g z µ, z µ ) = 1 2. We dene g(ψ,ψ) to be the probability density of the lifeted state function Ψ. Since a state function and its lift have the same probability, we may study quantum mechanics with lifted state functions.

17 The Probability of a Lifted State Recall that ( g z µ, z µ So, we obtain ) ( = g z µ, z µ ψ 2 = g(ψ,ψ). ) ( = 0, g z µ, z µ ) = 1 2. We dene g(ψ,ψ) to be the probability density of the lifeted state function Ψ. Since a state function and its lift have the same probability, we may study quantum mechanics with lifted state functions.

18 Hermitian Structure h on the Holomorphic Tangent Bundle T + (C n ) Hermitian structure always exists on the holomorphic tangent bundle T + (C n ). Theorem For each p C n, dene h p : T + p (C n ) T + p (C n ) C by h p (u, v) = g p (u, v) for u, v T + p (C n ). Then h is a Hermitian structure on T + (C n ).

19 Hermitian Structure h on the Holomorphic Tangent Bundle T + (C n ) Hermitian structure always exists on the holomorphic tangent bundle T + (C n ). Theorem For each p C n, dene h p : T + p (C n ) T + p (C n ) C by h p (u, v) = g p (u, v) for u, v T + p (C n ). Then h is a Hermitian structure on T + (C n ).

20 T + (C n ) and Lifted States Ψ Ψ = g(ψ,ψ) = g(ψ +,Ψ + ) = h(ψ +,Ψ + ) where Ψ + is the holomorphic part Ψ + = ψ µ z µ of the lifted state Ψ. So without loss of generality, we may consider Ψ + : R 3+1 T + (C n ) as a lifted state function.

21 T + (C n ) and Lifted States Ψ Ψ = g(ψ,ψ) = g(ψ +,Ψ + ) = h(ψ +,Ψ + ) where Ψ + is the holomorphic part Ψ + = ψ µ z µ of the lifted state Ψ. So without loss of generality, we may consider Ψ + : R 3+1 T + (C n ) as a lifted state function.

22 Connection Lifted Quantum Mechanics For an obvious reason, we would like to dierentiate sections (elds). If we cannot dierentiate them, we cannot do physics. Dierentiating sections of a bundle can be done by introducing a connection. In general, connection is not unique i.e. there is no unique way to dierentiate sections and one needs to make a choice of connection.

23 Connection Lifted Quantum Mechanics For an obvious reason, we would like to dierentiate sections (elds). If we cannot dierentiate them, we cannot do physics. Dierentiating sections of a bundle can be done by introducing a connection. In general, connection is not unique i.e. there is no unique way to dierentiate sections and one needs to make a choice of connection.

24 Connection Lifted Quantum Mechanics For an obvious reason, we would like to dierentiate sections (elds). If we cannot dierentiate them, we cannot do physics. Dierentiating sections of a bundle can be done by introducing a connection. In general, connection is not unique i.e. there is no unique way to dierentiate sections and one needs to make a choice of connection.

25 Hermitian Connection Theorem Let M be a Hermitian manifold. Given a holomorphic vector bundle π : E M and a Hermitian structure h, there exists a unique Hermitian connection. Denition A set of sections {e 1, e n } is called a unitary frame if h(e µ, e ν ) = δ µν. Associated with a tangent bundle TM over a manifold M is a principal bundle called the frame bundle LM = p M L pm, where L p M is the set of frames at p M. The structure group of the frame bundle LM is U(n) or SU(n) (if it is an oriented frame bundle).

26 Hermitian Connection Theorem Let M be a Hermitian manifold. Given a holomorphic vector bundle π : E M and a Hermitian structure h, there exists a unique Hermitian connection. Denition A set of sections {e 1, e n } is called a unitary frame if h(e µ, e ν ) = δ µν. Associated with a tangent bundle TM over a manifold M is a principal bundle called the frame bundle LM = p M L pm, where L p M is the set of frames at p M. The structure group of the frame bundle LM is U(n) or SU(n) (if it is an oriented frame bundle).

27 Hermitian Connection Theorem Let M be a Hermitian manifold. Given a holomorphic vector bundle π : E M and a Hermitian structure h, there exists a unique Hermitian connection. Denition A set of sections {e 1, e n } is called a unitary frame if h(e µ, e ν ) = δ µν. Associated with a tangent bundle TM over a manifold M is a principal bundle called the frame bundle LM = p M L pm, where L p M is the set of frames at p M. The structure group of the frame bundle LM is U(n) or SU(n) (if it is an oriented frame bundle).

28 Hermitian Connection Continued Let {e 1,, e n } be a unitary frame. Dene a local connection one-form ω = (ω ν µ ) by e µ = ω ν µ e ν. Theorem 2 e µ = e µ = Fµ ν e ν. Fµ ν are the coecients of the curvature 2-form F of the Hermitian connection or physically eld strength which is dened by F = dω ω ω.

29 Hermitian Connection Continued Let {e 1,, e n } be a unitary frame. Dene a local connection one-form ω = (ω ν µ ) by e µ = ω ν µ e ν. Theorem 2 e µ = e µ = Fµ ν e ν. Fµ ν are the coecients of the curvature 2-form F of the Hermitian connection or physically eld strength which is dened by F = dω ω ω.

30 Hermitian Connection Continued Let {e 1,, e n } be a unitary frame. Dene a local connection one-form ω = (ω ν µ ) by e µ = ω ν µ e ν. Theorem 2 e µ = e µ = Fµ ν e ν. Fµ ν are the coecients of the curvature 2-form F of the Hermitian connection or physically eld strength which is dened by F = dω ω ω.

31 Hermitian Connection Continued Let us dierentiate φ + with the Hermitian connection. ( φ + = z µ z µ ) = dz µ z µ + z µ ( ) z µ = dz µ z µ + z µ ωµ ν z ν. This allows us to dene a covariant derivative + for lifted state functions Ψ +. Denition + Ψ + = dψ µ z µ + ψ µ ω ν µ z ν.

32 Hermitian Connection Continued Let us dierentiate φ + with the Hermitian connection. ( φ + = z µ z µ ) = dz µ z µ + z µ ( ) z µ = dz µ z µ + z µ ωµ ν z ν. This allows us to dene a covariant derivative + for lifted state functions Ψ +. Denition + Ψ + = dψ µ z µ + ψ µ ω ν µ z ν.

33 Hermitian Connection Continued Let us dierentiate φ + with the Hermitian connection. ( φ + = z µ z µ ) = dz µ z µ + z µ ( ) z µ = dz µ z µ + z µ ωµ ν z ν. This allows us to dene a covariant derivative + for lifted state functions Ψ +. Denition + Ψ + = dψ µ z µ + ψ µ ω ν µ z ν.

34 Quantum Mechanics of a Charged Particle in an Electromagnetic Field Assume that ω u(1) = so(2). Then in terms of space-time coordinates (t, x 1, x 2, x 3 ), ω can be written as ω = iē h ρdt iē h A αdx α, α = 1,2,3. The covariant derivative + ψ + of the lifted state function Ψ + : R 3+1 C is then + Ψ + = dψ z + ωψ ( z = t iē h ρ ) ψ z dt + ( x α iē h A α ) ψ z dx α.

35 Quantum Mechanics of a Charged Particle in an Electromagnetic Field Assume that ω u(1) = so(2). Then in terms of space-time coordinates (t, x 1, x 2, x 3 ), ω can be written as ω = iē h ρdt iē h A αdx α, α = 1,2,3. The covariant derivative + ψ + of the lifted state function Ψ + : R 3+1 C is then + Ψ + = dψ z + ωψ ( z = t iē h ρ ) ψ z dt + ( x α iē h A α ) ψ z dx α.

36 Quantum Mechanics of a Charged Particle in an Electromagnetic Field Continued i h ( + Ψ + = ) ( ) i h t iē h ρ ψ z dt i h iē x α h A α ψ z dx α may be regarded as the four-momentum operator for lifted state functions Ψ + : R 3+1 T + (C). ( ) ( ) Let 0 = t iē h ρ z, α = iē x α h A α z, α = 1,2,3. Then the Schrödinger equation for a charge particle in an electromagnetic eld is given by where i hd 0 ψ = h2 2m D2 αψ + V ψ, D 0 = π 0 = t iē h ρ, D α = π α = x α iē h A α, α = 1,2,3.

37 Quantum Mechanics of a Charged Particle in an Electromagnetic Field Continued i h ( + Ψ + = ) ( ) i h t iē h ρ ψ z dt i h iē x α h A α ψ z dx α may be regarded as the four-momentum operator for lifted state functions Ψ + : R 3+1 T + (C). ( ) ( ) Let 0 = t iē h ρ z, α = iē x α h A α z, α = 1,2,3. Then the Schrödinger equation for a charge particle in an electromagnetic eld is given by where i hd 0 ψ = h2 2m D2 αψ + V ψ, D 0 = π 0 = t iē h ρ, D α = π α = x α iē h A α, α = 1,2,3.

38 Further Research Lifted Quantum Mechanics Non-relativistic equations for particles with higher spin? In particular, non-relativistic equation for spin-3/2 particle. Generalization to relativistic case?

39 Further Research Lifted Quantum Mechanics Non-relativistic equations for particles with higher spin? In particular, non-relativistic equation for spin-3/2 particle. Generalization to relativistic case?

40 Questions? Lifted Quantum Mechanics Thank you.