On complexified quantum mechanics and space-time Dorje C. Brody Mathematical Sciences Brunel University, Uxbridge UB8 3PH dorje.brody@brunel.ac.uk Quantum Physics with Non-Hermitian Operators Dresden: 5-25 June 20 - -
On complexified quantum mechanics - 2-2 June 20 Work based on: D. C. Brody & E. M. Graefe (20) On complexified mechanics and coquaternions Journal of Physics A44, 07200. D. C. Brody & E. M. Graefe (20) Six-dimensional space-time from quaternionic quantum mechanics (arxiv:05.3604). D. C. Brody & E. M. Graefe (20) Coquaternionic quantum dynamics for two-level systems Acta Polytechnica 5 (to appear; arxiv:05.4038).
On complexified quantum mechanics - 3-2 June 20 Quaternions and coquaternions Quaternions (Hamilton, 844) i 2 = j 2 = k 2 = ijk = and the cyclic relation ij = ji = k, jk = kj = i, ki = ik = j. The squared norm of q = q 0 + iq + jq 2 + kq 3 is q 2 = q0 2 + q 2 + q2 2 + q3. 2 Coquaternion (Cockle, 849) i 2 =, j 2 = k 2 = ijk = + and the skew-cyclic relation ij = ji = k, jk = kj = i, ki = ik = j. The squared norm of q = q 0 + iq + jq 2 + kq 3 is q 2 = q0 2 + q 2 q2 2 q3. 2
On complexified quantum mechanics - 4-2 June 20 Complex formulation of real mechanics Hamilton s equations: ṗ = H x, ẋ = H p. Introduce complex phase-space variable z = 2 (x + ip). The canonical equation of motion then reads i dz dt = H z, which is just the Schrödinger equation (Dirac, 927): i d z dt = H z, H = z Ĥ z z z.
On complexified quantum mechanics - 5-2 June 20 Complexification Recall that equations of motions are: ṗ = H x, ẋ = H p ; idz dt = H z. Complexification (i) x x 0 + ix, p p 0 + ip ; z 2 [(x 0 p ) + i(x + p 0 )]? Complexification (ii) x x 0 + jx, p p 0 + jp ; z 2 [x 0 + ip 0 + jx + kp ] Here, i,j, k can be unit quaternions or coquaternions. Quaternions Quaternionic quantum mechanics Coquaternions PT-symmetric quantum mechanics
On complexified quantum mechanics - 6-2 June 20 Quaternionic/coquaternionic quantum dynamics One-parameter unitary group Û t is generated by a dynamical equation: where Ψ = iĥ Ψ, Ĥ is Hermitian, i is skew-hermitian unitary, and both commute with Ût. We let Ĥ be given arbitrarily, and restrict i to be the unique unit imaginary (co)quaternion that commute with Ĥ.
On complexified quantum mechanics - 7-2 June 20 Quaternionic spin- 2 particle A generic 2 2 Hermitian Hamiltonian can be expressed in the form 5 Ĥ = u 0 + u lˆσ l, where {u l } R, and {ˆσ l are the quaternionic Pauli matrices: ( ) ( ) ( ) 0 0 i 0 ˆσ =, ˆσ 0 2 =, ˆσ i 0 3 =, 0 ( ) ( ) 0 j 0 k ˆσ 4 =, ˆσ j 0 5 =. k 0 Having specified the Hamiltonian, we must select a unit imaginary quaternion such that the evolution operator Û t = exp( iĥt) is unitary. This is given by where ν = u 2 2 + u2 4 + u2 5. l= i = ν (iu 2 + ju 4 + ku 5 ),
On complexified quantum mechanics - 8-2 June 20 Quaternionic spin dynamics To determine the dynamics we introduce a quaternionic Bloch vector: σ l = Ψ ˆσ l Ψ / Ψ Ψ, l =,...,5. Then for each component we can work out the dynamics by making use of the Schrödinger equation. After rearrangements we deduce that 2 σ = νσ 3 u 3 (u 2 σ 2 + u 4 σ 4 + u 5 σ 5 )/ν 2 σ 2 = (u 2 u 3 σ u u 2 σ 3 + u 0 u 5 σ 4 u 0 u 4 σ 5 )/ν 2 σ 3 = νσ + u (u 2 σ 2 + u 4 σ 4 + u 5 σ 5 )/ν 2 σ 4 = (u 3 u 4 σ u 0 u 5 σ 2 u u 4 σ 3 + u 0 u 2 σ 5 )/ν 2 σ 5 = (u 3 u 5 σ + u 0 u 4 σ 2 u u 5 σ 3 u 0 u 2 σ 4 )/ν. These evolution equations preserve the normalisation condition: σ 2 + σ 2 2 + σ 2 3 + σ 2 4 + σ 2 5 =, which is the defining equation for the state space S 4 R 5. SO(5) symmetry.
On complexified quantum mechanics - 9-2 June 20 Dimensional reduction As in any physical theory modelled on a higher-dimensional space-time, it is important to identify in which way dimensional reduction occurs. For this purpose, let us define the three spin variables: σ x = σ, σ y = ν (u 2σ 2 + u 4 σ 4 + u 5 σ 5 ), σ z = σ 3. Then a calculation making use of the generalised Bloch dynamics shows that 2 σ x = νσ z u 3 σ y 2 σ y = u 3 σ x u σ z 2 σ z = u σ y νσ x. The reduced spin dynamics is thus confined to the state space where r is time independent. σ 2 x + σ 2 y + σ 2 z = r 2, What about the motion of the internal variables of σ y : σ 2, σ 4, and σ 5?
On complexified quantum mechanics - 0-2 June 20 The motion of these variables lies on a cylinder in R 3 : (u 2 σ 4 u 4 σ 2 ) 2 + (u 4 σ 5 u 5 σ 4 ) 2 + (u 5 σ 2 u 2 σ 5 ) 2 = ν 2 c 2, where c 2 = r 2 is the squared radius of the cylinder. Figure : Dynamical trajectories on the cylindrical subspace.
On complexified quantum mechanics - - 2 June 20 Figure 2: Bloch dynamics on the reduced state space. Symmetry breaking: SO(5) SO(3) U().
On complexified quantum mechanics - 2-2 June 20 Observable effect of extra dimensions (i) Spin-measurement statistics gives: σx 2 + σy 2 + σz 2 = c 2 (ii) Interference Peres 979; Adler 988; Adler & Anandan 996 (iii) Superconductor-antiferromagnet Zhang 997, 998, 2000
On complexified quantum mechanics - 3-2 June 20 Coquaternionic spin- 2 particle A generic 2 2 Hermitian Hamiltonian can be expressed in the form 5 Ĥ = u 0 + u lˆσ l, where {u l } R, and {ˆσ l are the coquaternionic Pauli matrices: ( ) ( ) ( ) 0 0 i 0 ˆσ =, ˆσ 0 2 =, ˆσ i 0 3 =, 0 ( ) ( ) 0 j 0 k ˆσ 4 =, ˆσ j 0 5 =. k 0 The evolution operator Û t = exp( iĥt) is unitary if we set l= i = ν (iu 2 + ju 4 + ku 5 ), where ν = u 2 2 u2 4 u2 5 or ν = u 2 4 + u2 5 u2 2. The eigenvalues of H are E ± = u 0 ± u 2 + u2 2 + u2 3 u2 4 u2 5.
On complexified quantum mechanics - 4-2 June 20 Polar decomposition of coquaternions Let q = q 0 + iq + jq 2 + kq 3 be a generic coquaternion. If qq > 0 and q 2 q 2 2 q 2 3 > 0, then where i q = iq + jq 2 + kq 3 q 2 q2 2 q2 3 If qq > 0 but q 2 q 2 2 q 2 3 < 0, then where i q = iq + jq 2 + kq 3 q 2 + q2 2 + q2 3 q = q e i qθ q = q (cosθ q + i q sinθ q ), and θ q = tan ( q 2 q 2 2 q2 3 q 0 q = q e i qθ q = q (coshθ q + i q sinhθ q ), ) and θ q = tanh ( q 2 + q 2 2 + q2 3 q 0 If qq > 0 and q 2 q 2 2 q 2 3 = 0, then q = q 0 ( + i q ), where i q = (iq + jq 2 + kq 3 )/q 0.. ).
On complexified quantum mechanics - 5-2 June 20 Finally, if qq < 0, then where i q = iq + jq 2 + kq 3 q 2 + q2 2 + q2 3 q = q e i qθ q = q (sinhθ q + i q coshθ q ), and θ q = tanh ( q 2 + q 2 2 + q2 3 q 0 ).
On complexified quantum mechanics - 6-2 June 20 Coquaternionic spin dynamics In the case of a coquaternionic spin- particle, the generalised Bloch equations 2 are: 2 σ = νσ 3 u 3 ν (u 2σ 2 + u 4 σ 4 + u 5 σ 5 ) 2 σ 2 = ν (u 2u 3 σ u u 2 σ 3 + u 0 u 5 σ 4 u 0 u 4 σ 5 ) 2 σ 3 = νσ + u ν (u 2σ 2 + u 4 σ 4 + u 5 σ 5 ) 2 σ 4 = ν ( u 3u 4 σ + u 0 u 5 σ 2 + u u 4 σ 3 + u 0 u 2 σ 5 ) 2 σ 5 = ν ( u 3u 5 σ u 0 u 4 σ 2 + u u 5 σ 3 u 0 u 2 σ 4 ), These evolution equations preserve the normalisation condition: σ 2 + σ 2 2 + σ 2 3 σ 2 4 σ 2 5 =, which is the defining equation for the hyperbolic state space. SO(3,2) symmetry
On complexified quantum mechanics - 7-2 June 20 Dimensional reduction: Hermitian case As before, we define the three reduced spin variables σ x = σ, σ y = ν (u 2σ 2 + u 4 σ 4 + u 5 σ 5 ), σ z = σ 3. Then a short calculation shows that when u 2 4 + u 2 5 < u 2 2. This preserves Standard unitary dynamics. 2 σ x = νσ z u 3 σ y 2 σ y = u 3 σ x u σ z 2 σ z = u σ y νσ x, σ 2 x + σ 2 y + σ 2 z = r 2.
On complexified quantum mechanics - 8-2 June 20 Figure 3: Reduced Bloch dynamics for a coquaternionic spin- 2 particle. Dimensional reduction: non-hermitian, real energy case When u 2 4 + u 2 5 > u 2 2, we have 2 σ x = νσ z u 3 σ y 2 σ y = u 3 σ x + u σ z 2 σ z = u σ y + νσ x.
On complexified quantum mechanics - 9-2 June 20 Unitary dynamics on a hyperboloid: σ 2 x σ 2 y + σ 2 z = r 2. Figure 4: Reduced dynamics for a coquaternionic spin- 2 particle with unbroken symmetry.
On complexified quantum mechanics - 20-2 June 20 Dimensional reduction: non-hermitian, real energy case When u 2 4 + u 2 5 > u 2 2, and u 2 + u 2 2 + u 2 3 > u 2 4 + u 2 5 so that energy eigenvalues form complex conjugate pairs, the orbits close on a hyperboloid. Figure 5: Reduced dynamics for a coquaternionic spin- 2 particle with broken symmetry.
On complexified quantum mechanics - 2-2 June 20 Figure 6: Reduced dynamics for a coquaternionic spin- 2 particle with broken symmetry.
On complexified quantum mechanics - 22-2 June 20 Closing remarks Symplectization, Complexification and Mathematical Trinities V. I. Arnold (997) Maybe there is some complexified version of the quantum Hall effect, the three dimensional transversal being replaced by a five dimensional one. It would have been easy to predict the quantum Hall effect and the Berry phase theory simply by complexifying the theory of monodromy of quadratic forms from the Modes and Quasimodes. This opportunity was lost. We may also miss more opportunities not studying the quaternionic version of the modes and quasimodes theory.
On complexified quantum mechanics - 23-2 June 20 Open questions Relation to the C operator Mapping from the hyperbolic state space to the spherical state space Infinite-dimensional case and boundary conditions