Clifford Algebra
Note 5
TOMONAGA’s Super Multi-time Theory
TANAKA Akio
1 <Schrödinger equation>
State vector ψ
Time t
Electromagnetic field A
Hamiltonian H
i
ψ(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
-
em
Electromagnetic 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
-
em
t }
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
( x
1
,
t
1
; … ; x
N
,
t
N
) = 0 (4)
5 <Tomonaga’s representation of electromagnetic field>
Unitary transformation
U (t) = exp {
(H
1
+ H
2
) t }
Schrödinger equation
[H
1
+ H
2
+ H
12
+
] ψ(t) = 0
Independent time variant t
xyz
at each point in space
[ H
12
(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 vector
Φ
[C] = Φ[T
xyz
] Φ[C]
[ H
12
( P ) +
] Φ[C] = 0 (6)
[References]
<Past work on multi-time themes>
Aurora Theory / Dictoron, Time and Symmetry <Language Multi-Time Conjecture> / Tokyo October 6,
2006
Aurora Theory / Word, Phrase and Sentence <Language Multi-Time Conjecture 2> / Tokyo October 25,
2006
Aurora Theory / Distance and Time <Language Multi-Time Conjecture 3> 5
th
Time for KARCEVSKIJ /
Tokyo October 28, 2006
Language and Spacetime / Time Flow in Word For KOBARI Akihiro and His Time / Tokyo May 3, 2007
<For more details>
Invitation by Theme-Time / Data Arranged at Tokyo January 6, 2008
Aurora Theory
Language and Spacetime
Tokyo January 25, 2008
Sekinan Research Field of Language
www.sekinan.org