ColiffordAlgebra

Note 5

TOMONAGA’s Super Multi-time Theory

TANAKA Akio

1 <Schrödinger equation>

State vector ψ

Time t

Electromagnetic field A

Hamiltonian H

iψ(t) = Hψ(t), ψ(0) = ψ (1)

2 <Dirac’s paraphrase of Schrödinger equation >

Coordinate x

Momentum p

Electron N in number

Electromagnetic field A

H^-_emElectromagnetic field Hamiltonian

[ H^-_em+ H_n ( x_n, p_n, A (x_n) ) + ] ψ(t) = 0 (2)

3 <Representation by unitary transformation>

u(t) = exp{ H^-_emt }

A (x_n, t) = u(t) A (x_n) u(t)^-1

Φ(t) = u(t) ψ(t)

[ H_n ( x_n, p_n, A (x_n, t) ) + ] Φ(t) = 0 (3)

4 < Dirac’s multi-time theory- Time variant in number N >

[H_n ( x_n, p_n, A (x_n, t_n) ) + ] Φ( x₁,t₁; … ; x_N,t_N) = 0 (4)

5 <Tomonaga’s representation of electromagnetic field>

Unitary transformation

U (t) = exp { (H₁ + H₂ ) t }

Schrödinger equation

[H₁+ H₂ + H₁₂+ ] ψ(t) = 0

Independent time variant t_xyzat each point in space

[ H₁₂ (x, y, z, t_xyz ) + ] Φ(t) = 0 (5)

6 < Tomonaga’s super multi-time theory>

Super curved surface C

Point on C P

4-dimensional volume’s transformation of C C_P

Infinite small variation of state vectorΦ[C] = Φ[T_xyz] Φ[C]

[ H₁₂ ( P ) + ] Φ[C] = 0 (6)

[References]