Quantum, Classical and Symmetry Aspects of Max Born Reciprocity
L.M. Tomilchik
B.I. Stepanov Institute of Physics,
National Academy of Sciences of Belarus
Minsk, 16–19 May 2017
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Topics
1 Reciprocal Symmetry and Maximum Tension Principle 2 Maximum Force and Newton gravity 3 Extended Phase Space (QTPH) as a Basic Manifold 4 Complex Lorentz group with Real Metric as Group of
Reciprocal Symmetry
5 One-Particle Quasi-Newtonian Reciprocal-Invariant
Hamiltonian dynamics
6 Canonic Quantization: Dirac Oscillator as Model of
Fermion with a Plank mass
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Born Reciprocity version
1 Reciprocity Transformations (RT):
- xµ
- pµ
- pµ
- xµ
qe
→
,
→ −
.
- qe
- pe
- pe
2 Lorentz and Reciprocal Invariant quadratic form:
1
qe2
1
pe2
SB2
=
xµxµ +
pµpµ, ηµν = diag{1, −1, −1, −1}
3 xµ, pµ are quantum-mechanical canonic operators:
[xµ, pν] = i~ηµν
4 Phenomenological parameter according definition pe = Mc so that
qe = ~/Mc = Λc (Compton length).
The mass M is a free model parameter.
5 Two dimensional constants qe[length], pe[momentum] connected by corellation:
qepe = ~ (Plank constant).
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Relativistic Oscillator Equation (ROE)
Selfreciprocal-Invariant Quantum mechanical Equation:
- ꢀ
- ꢁ
2
∂
−
+ ξµξµ Ψ(ξ) = λBΨ(ξ),
∂ξµ∂ξµ
where ξµ = xµ/Λc. The (ROE) symmetry ≈ U(3, 1). There are solution corresponding linear discrete mass spectrum.
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Maximum Tension Principle (MTP)
The space-time and momentum variables are to be just the same dimentions of a quantity. It is necessary to have the universal constant with dimention: momentum/length, or equivalently: mass/time, energy/time, momentum/time. In reality: MTP [Gibbons, 2002] Gibbons Limit:
ꢀꢀ
ꢁ
dp dt
c4
Maximum force Fmax
==
- =
- = FG,
4GN
max
ꢁ
dE
c5
Maximum power Pmax
- =
- = PG.
dt
4GN
max
[momentum]/[length] – Universal constant
pe qe
c3
- æ0 =
- =
[LMT, 1974, 2003]
4GN
- −1
- −2
It is evident: æ0 = c
F
= c
P
. All these values are
- max
- max
essentially classic (does not contain Plank constant ~).
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Born Reciprocity and Gibbons Limit
Let the parameter a to be some fixed (nonzero!) value of action S. We replace initial Born’s relation
qepe = ~ by πqepe = a
without beforehand action discretness supposition. We demand
the nonzero area Sfix = πqepe of the phase plane only. The second suggestion is:
pe qe
c3
- æ0 =
- =
(universal constant).
4GN
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Now the parameters qe, pe are defined separately
- r
- r
a
aæ
qe =
,
pe =
πæ
π
Evidently: the phenomenological constants Ee = pec [Energy] and te = c−1qe [time] satisfy the conditions
Ee te
c5
Eete = πa,
= c2æ =
4GN
Under Born’s parametrization
2MGN pe = Mc, qe =
= rg(M) – gravitational radius.
c2
General mass-action correlation
ac
.
4πGN
M2 =
It is valid as in classical as in quantum case. Under supposition when amin = h we receive
s
hc
12
~c
GN
M2 =
=
MP2 l
,
MP l
=
– Plank mass.
4πGN
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Maximum ꢂFꢂorce and Modified Newton Gravity
mM
r2
c4 2mGN 2MGN
1
r2
ꢂ ꢂ
(NG):
f = GN
≡
4GN
- c2
- c2
p
rgRg
r2 r02
= FG
≡ FG
(r > 0, r0 = rgRg)
r2 r02
G r2
ꢂꢂꢂ
- ꢂ
- ꢂ
ꢂꢂ
ꢂꢂꢂ
max
ꢂ
(ModNG):
fmod = F
(r > r0),
fmod
= FG
ꢂ
- N
- N
r=r0
The corresponding gravitational potential Energy
r02
mod gr
U
= −FG
r
possess a minimum under r = r0
c4 2GN
c2
- √
- √
Ugmrod(min) = −FGr0 = −
mM = − mM
4GN c2
2
We have dealing (instead of point-like masses) with a pair spacely extended simple massive gravitating objects like the elastic balls or drops with a high surface tension.
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r0
- M
- m
- Rg
- rg
√
12
Erest(M, m) = Mc2 + mc2 − mMc2
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The whole energy such a “binary” system is as follows:
√
1
E(M, m) = Mc2 + mc2 − mMc2
2
ꢃr
ꢄ
12
µ
= (M + m)c2 1 −
,
M + m
where µ = mM/(m + M).
Mc2, mc2, (M + m)c2 – rest energy
p
√
12
12
- ∆E =
- µ(M + m)c2 =
mMc2 – energy corresponding
√
12
the “mass defect” ∆m =
mM
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“Gravitational fusion” picture
[eliminated energy]
∆E
(M + m)c2
λ =
=1
[rest energy]
- √
- √
- mM
- δ
- =
- =
,
- 2(M + m)
- 2 1 + δ
where
m
Mmin Mmax
- δ =
- ,
δ0 6 δ 6 1, δ0 =
M
∆Eeliminated
14
Maximum λmax
- =
- =
Erest
under δ = 1 (m = M)
Such a “fusion” of two equal rest mass M lead to mass = 3M/2, the Energy ∆E = Mc2/2 eliminate (3 : 1).
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Extended Phase Space as a Basic Manifold
Space of states (QTP) and Energy (H) (sec [J. Synge, 1962]) in general: 2 + 2N dimensions. The metric is
N
- }|
- z
- {
ηAB = diag{1, −1, −1, . . . , −1}
(X, Y ) – pair of conjugated pseudoeuclidean vectors XA, YA,
A, B = 0, 1, . . . , N (N + 1 dimention).
The Poisson brackets for ϕ(X, Y ), f(X, Y ),
- ꢃ
- ꢄ
N
X
∂ϕ ∂f
∂Xk ∂Yk
∂ϕ ∂f
∂Y k ∂Xk
{ϕ, f}A,B
=
−
k=0
so that
{XA, YA}A,B = ηAB
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Dynamic System in (QT, P H) is defined, when the Energy Surface (2N + 1 dimention)
Ω(X, Y ) = 0
is defined. Solving this equation on Y0 we obtain Energy
equation
Y0 = F (Xk, Yl, X0) k, l = 1, 2, . . . , N
The corresponding Hamilton function of the System is by def:
H ≡ Y0 = H (Xk, Yl, Y0)
We shall consider (2 + 2 · 3) – dimensional version (QT, P H)
A, B → µ, ν = 0, 1, 2, 3, ηµν = diag{1, −1, −1, −1},
Xµ, Xµ are real 4-dimensional pseudoeuclidean SO(3, 1) – vectors.
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Complexification of (QT, P H)
Let us introduce the complex 4-vector Zµ according definition:
Zµ = Xµ + iY µ, ηµν = diag{1, −1, −1, −1}
and define the real norm
ZµZµ∗ = ZµηµνZµ∗
- 2
- 2
- 2
- 2
= |Z0| − |Z1| − |Z2| − |Z3| = inv
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Complex Lorentz Group with a real metric
(Barut Group [Barut 1964])
The group of 4 × 4 complex matrices Λ of transformations
Z0 = ΛZ, Z = X + iY
satisfying the condition
ΛηΛ† = η, († – hermitien conjugation)
¯
Notice: the diagonal matrices ΛR and ΛR ∈ (BG)
ΛR = −iI4 = −idiag{1, 1, 1, 1}
and
- ¯
- ΛR = −iηµν = −idiag{1, −1, −1, −1}
Metric invariant (no positive-defined)
- 2
- 2
- 2
- 2
ZµZµ∗ = |Z0| − |Z1| − |Z2| − |Z3| = inv
2
2
Three possibilities: |Z| ≷ 0, |Z| = 0 – isotropic case
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SU(3, 1) – subgroup structure
(QTPH)
SOvH
QT = {r, ct} = xµ
Lk
ꢅ
P H = p, E/c = pµ
~βv = ~v/c
C
center
*
C4
SOH
Ukl
SUH
(PTQH)
ꢅ
P T = p, F0t = Πµ
Mk
QH = {r, E/F0} = Ξµ
SOf H
- ~
- ~
βf = f/F0
F0 = Fmax
k, l = 1, 2, 3
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Reciprocal Invariant SB2 as BG metric invariant
ZµZµ∗
By def: Zµ = Xµ + iYµ
- 1
- 1
- 1
- 1
- Xµ =
- xµ =
{x0, xk}, Yµ =
pµ =
{p0, pk}.
- qe
- qe
- pe
- pe
Then:
- !
ꢁ
- !
- 3
- 3
- X
- X
1
qe2
1
pe2
SB2
=
x20 − xk2
+
p20 − pk2
=
- k=1
- k=1
ꢀ
- 3
- 3
x02 qe2 p02 pe2 xk2 qe2 pk2 pe2
- X
- X
- 2
- 2
- =
- +
−
+
- = |Z0| −
- |Zk| = ZµZµ∗
- k=1
- k=1
Let xµ, pµ to be 4-vectors under SOv(3, 1)-transformations
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Notice! There exist alternative:
¯
Zµ = Xµ + iYµ
- ¯
- ¯
where
- ꢃ
- ꢄ
- ꢃ
- ꢄ
- x0 pk
- p0 xk
,
pe qe
def
=
def
=
- ¯
- ¯