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Quantum, Classical and Symmetry Aspects of Max Born Reciprocity

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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

1 / 36

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

2 / 36

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).

3 / 36

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.

4 / 36

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 ~).

5 / 36

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

6 / 36

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

7 / 36

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.

8 / 36

r0

  • M
  • m

  • Rg
  • rg

√

12

Erest(M, m) = Mc2 + mc2 − mMc2

9 / 36

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

10 / 36

“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).

11 / 36

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

12 / 36

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.

13 / 36

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

14 / 36

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

15 / 36

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

16 / 36

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

17 / 36

Notice! There exist alternative:

¯

Zµ = Xµ + iYµ

  • ¯
  • ¯

where

  • ꢃ
  • ꢄ
  • ꢃ
  • ꢄ

  • x0 pk
  • p0 xk

,

pe qe

def

=

def

=

  • ¯
  • ¯

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  • Cell Theory of Nature

    Cell Theory of Nature

    CELL THEORY OF NATURE Work in process David Ritz Finkelstein1 October 17, 2008 1Physics, Georgia Institute of Technology, Atlanta, Georgia. fi[email protected] 2 Time is the number of motion with regard to before and after. Aristotle, Physics I would indeed admit these infinitely small spaces and times in geometry, for the sake of invention, even if they are imaginary. But I am not sure whether they can be admitted in nature. G. W. Leibniz [3] It is, for example, true that the result of two successive acts is unaffected by the order in which they are performed; and there are at least two other laws which will be pointed out in the proper place. These will perhaps to some appear so obvious as to be ranked among necessary truths, and so little important as to be undeserving of special notice. And probably they are noticed for the first time in this Essay. Yet it may with confidence be asserted, that if they were other than they are, the entire mechanism of reasoning, nay the very laws and constitution of the human intellect, would be vitally changed. A Logic might indeed exist, but it would no longer be the Logic we possess. G. Boole [14] To be sure, it has been pointed out that the introduction of a space-time contin- uum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is, to the elimination of continuous functions from physics.