Thursday, 13 June 2013

TOMONAGA’s Super Multi-time Theory


Note 5
TOMONAGA’s Super Multi-time Theory


1 <Schrödinger equation>
State vector     ψ
Time     t
Electromagnetic field     A
Hamiltonian     H
iψ(t) = (t),   ψ(0) = ψ     (1)
2 <Dirac’s paraphrase of Schrödinger equation >
Coordinate     x
Momentum     p
Electron     N in number
Electromagnetic field     A
H-em     Electromagnetic field Hamiltonian
H-em Hn ( xnpn(xn) ) +   ] ψ(t) = 0     (2)
3 <Representation by unitary transformation>
u(t) = exp{ H-em}
(xnt) u(t) A (xn) u(t)-1
Φ(t) = u(t) ψ(t)
Hn ( xnpn(xnt) ) +   ] Φ(t) = 0     (3)
4 < Dirac’s multi-time theory- Time variant in number >
[Hn ( xnpn(xntn) ) +   ] Φx1, t1; … ; xN, t) = 0     (4)
5 <Tomonaga’s representation of electromagnetic field>
Unitary transformation
U (t) = exp {  (H1 + H2 ) t }    
Schrödinger equation
[HH2 + H12  ] ψ(t) = 0    
Independent time variant txyz at each point in space 
H12 (xyztxyz ) +   ] 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     CP
Infinite small variation of state vectorΦ[C] = Φ[Txyz]      Φ[C]
H12 ( P ) +   ] Φ[C] = 0     (6)

[References]
<Past work on multi-time themes>
<For more details>

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