QUANTUM DYNAMICS AND CONFORMAL GEOMETRY:

THE “AFFINE QUANTUM MECHANICS”

From Dirac’s equation to the EPR Quantum Nonlocality

Enrico SantamatoUniversity of Naples, Italy

Quantum Mechanics meets Gravity, La Sapienza, RomaDecember 2, 2011

Francesco De MartiniAccademia dei Lincei, Rome, Italy

AFFINE QUANTUM MECHANICS

According to Felix Klein “Erlangen Program” the “affine” geometrydeals with intrinsic geometric properties that remain unchangedunder “affine transformations” (affinities). These consist of collinearity transformations, e.g. sending parallels into parallelsand preseving ratios of distances along parallel lines.

The “conformal geometry” by Hermann Weyl is considered a kind of affine geometry preserving angles.

Cfr: H. Coxeter, Introduction to Geometry (Wiley, N.Y. 1969).

We aim at showing that the wave equation for

quantum spin (in particular Dirac’s spin ½ equation)may have room in the classical mechanics of the “relativistic

Top”

A.J.Hanson, T. Regge, Ann.Phys (NY) 87, 498 (1974); L.Shulman, Nucl.Phys B18, 595(1970).E. Sudarshan, N. Mukunda, Classical Dynamics, a modern perspective (Wiley, New York, 1974)

Niels Bohr, Wolfgang Pauliand the spinning Top

(Lund, 1951)

PROGRAM:

.1) DERIVATION OF DIRAC’s EQUATION

.2) WEYL’s CURVATURE AND THE ORIGIN OF QUANTUM NONLOCALITY

3) ELECTRON “ZITTERBEWEGUNG”

4) KALUZA-KLEIN –JORDAN THEORIES

5) ISOTOPIC SPIN

E.Santamato, F. De Martini , ArXiv 1107.3168v1 [quant-ph]

)(x

PTop’s trajectory:

Tetrad (a) (fourlegs, vierbein)

Tetrad vector = vector’s components on Tetrad 4-legs

)(ae

( ' )

( 1, 1, 1, 1); { , ; , :0, 1, 2, 3}

( / ) ; 6 : ( 1,....6)

: ' ( .)

a b ab

ab

a a

g e e g tetrad s normalization equation

g g diag a b

d e d e Euler angles

g top s angular velocity skewsymm

)..(: timepropergeparameter

LAGRANGIANS in

a: constant = (h/mc) Compton wl

is -parameter independent:

[H. Rund, The Hamilton-Jacobi equation (Krieger, NY; 1973)]

10V

10-D

span thedynamical configuration space:

Weyl’s affine connection and curvature:

:dim10

''

ensionsnDincalculatedbecanRcurvature

scalarsWeyloveralltheconnectionaffinesWeylBy

W

ijk

Hermann Weyl gauge - invariant Geometry

(*) H.Weyl, Ann. d. Physik, 59, 101 (1919).

In Riemann geometry: change of component of a contravariant vector under

“parallel transport” along

Scalar product of covariant - contravariant vector : leading to:

Then, covariant differentiation :

____________________________

In Weyl’s geometry :

":";: connectionaffinedxdx

2,

1

( ) ; 2 ;

" " : : " ' " !

: ; " " : ( )

/ ( ) ( ) /

i

i i i

l x l g g

New gauge vector field Weyl s potential

Weyl conformal transformation l e l local gauge field q

q grad q q q

.

dx ,

.0:; gtensormetricofationDifferenti

2 ( ) 0l

WEYL’s CONFORMAL MAPPING (CM) (*)

= spacetime dependent change of the unit of length L:

( )qds e ds

( ) dim si

: ( " " "dim "

( arg )).

, ( ) " "

W

W

Under CM any physical quantity with en ons L is assigned a trans

formation law X e X W weight or ensional number of the

object like the electric ch e in electrodynamics

Thus CM is a unit transformation amo

" ".

: : 2; : 2; : 2; : 0; : 2;

.

4 .

ik lik ik

unting to a space time redefinition

of the unit of length

Examples g W g W g W W R W

etc

Conformal Mapping preserves angles between vectors

(*) H. Weyl, Ann. der Physik, 59, 101 (1919); Time, Space, Matter (Dover, NY, 1975)

CONFORMAL GROUP :

.1) 6 – parameters Lie group, isomorphic to the proper, orthochronous, homogeneous Lorentz group.

.2) Preserves the angle between two curves in space time and its direction.

.3) In flat spacetime of Special Relativity the relevant group structure is the inhomogeneous Lorentz group (Poincare’ group).

In General Relativity , if spacetime is only “conformally flat” (i.e. Weyl’s conformal tensor: ) we obtain a larger group (15 parameters) of which the Poincaré group is a subgroup.

0C

1( )

21

( );6

C R R g R g R g R g

R g g g g R Riemann curvature tensor

Weyl’s conformal tensor:

P. Bergmann, Theory of Relativity, (Dover 1976, Pg. 250).

( )A A x

where the particle’s mass is replaced by the Weyl’s scalar curvature:

In place of L assume a conformally - invariant Lagrangian:

gqg i )(

Search for a family of equidistant hypersurfaces S = constant as bundles of extremals via Hamilton-Jacobi equation:

Nonlinear partial differential equations for the unknown S(q) and once the metric tensor is given. S(q) is the Hamilton’s “Principal function”.

)(q )(qgik

HAMILTON – JACOBI EQUATIONHAMILTON – JACOBI EQUATION

S(x)

S(x)

S(x) : Hamilton’s Principal Function ( )

i i

S xp

q

By the “ansatz” solution, with Weyl “weight”: W = (2 – n)/2

and, by fixing, for D=10 :

the “classical” Hamilton - Jacobi equation is linearized leading to:

_____________________________________________________________

In the absence of the e.m. field the above equation reduces to: where is the Laplace–Beltrami operator and

is the “conformal” Laplacian , i.e. a Laplace- de Rham operator.

,0, jA

L̂

./6 2aR

( )[ ] [ ( ) / ]

ˆ( ) (!)" "

i ii

S xp i i S x

x

Momentum operator p complex phase

STANDARD DEFINITION OF THE:

“HAMILTON’s PRINCIPAL FUNCTION” S(x) :

expressing the Weyl’s invariant current – density:

which can be written in the alternative, significant form:

______________________________________________

This is done by introducing the same “ansatz”:

The above results show that the scalar density is transported along

the particle’s trajectory in the configuration space, allowing a possible statistical

interpretation of the wavefunction according to Born’s quantum mechanical rule.

The quantum equation appears to be mathematically equivalent to the classical

Hamilton-Jacobi associated with the conformally – invariant Lagrangian

and the Born’s rule arises from the conformally invariant zero - divergence

current along any Hamiltonian bundle of trajectories in the configuration space.

It is possible to show that the Hamilton – Jacobi equation can account for the

quantum Spin-1/2.

2

: (3,1) (2)

[ (0, 1/ 2) (1/ 2,0)

:

( 1), ( 1)] ( , ) ( , )

/

( ) :

;

s

s s

s

Homomorphism SO SL

eigenvalues of the

Casimir operators

u u v v u v u v

Dotted undotted spinors

Space inversion I Parity

I I

I

{ 1, }

' var : 4 ' .s

i

Because of equation s I in iance Dimension Dirac s spinor

4 - D solution, invariant under Parity :

: (2u+1) (2v+1) matrices accounting for transf.s :

, : 2-component spinors accounting for space-time coord.s:x

4 components Dirac’s equation

Where:

By the replacement :

cancels, and by setting: the equation reproduces exactly the quantum – mechanical results given by:

L.D. Landau, E.M. Lifschitz, Relativistic Quantum Theory (Pergamon, NY, 1960)L.S. Schulman, Nucl. Phys. B18, 595 (1970).

the e.m. term:

THE SQUARE OF THE DIRAC’S EQUATION FOR THE SPIN ½

CAN BE CAST IN THE EQUIVALENT FORM:

0

WHERE THE GAMMA MATRICES OBEY TO THE CLIFFORD’s ALGEBRA:

THE 4 – MOMENTUM OPERATOR IS:

IN SUMMARY: our results suggest that:

.1) The methods of the classical Differential Geometry may be considered as an inspiring context in which the relevant paradigms of modern physics can be investigated satisfactorily by a direct , logical, (likely) “complete” theoretical approach.

.2) Quantum Mechanics may be thought of as a “gauge theory” based on “fields” and “potentials” arising in the context of diffe-rential geometry.Such as in the geometrical theories by: Kaluza, Klein, Heisenberg, Weyl, Jordan, Brans-Dicke, Nordström, Yilmaz, etc. etc.

.3) Viewed from the above quantum - geometrical perspective, “GRAVITATION” is a “monster” sitting just around the corner…….

QUANTUM MECHANICS : A WEYL’s GAUGE THEORY ?

LOOK AT THE DE BROGLIE – BOHM THEORY

Max Jammer,

The Philosophy ofQuantum Mechanics

(Wiley, 1974; Pag. 51)

DE BROGLIE - BOHM

(Max Jammer, The Philosophy of Quantum Mechanics, Wiley 1974; Pag. 52)

22

2

2

2

" " :

: exp( / )2

1 1[ dim : 3]

2 2 4

" " ' :

1 1( 1) .

4 2 4

[

k kk k

k kk k

W

De Broglie Bohm Quantum Potential

Q For wavefunction iS hm

Q In space nm

Overall Curvature in Weyl s geometry

nR R n

dim : 10 ]In spacetime n FULLY RELATIVISTIC THEORY

( )

' .

:

1) ( ' )

2) , . .

WQ R R Curvature due to

Weyl s gauge field

Physical effects of Weyl coupling among

particles via space time curvature

Quantum Interference Young s IF

Quantum Nonlocality etc etc

TOTAL SPACE TIME CURVATURE DUE TO AFFINE CONNECTION (i. e. TO CHRISTOFFEL SYMBOLS)IMPLIED BY 10-D METRIC TENSOR G

Einstein-Podolsky-Rosen “paradox” (EPR 1935)

1 2, )( i jqq1 2

“SPOOKY ACTION – AT – A – DISTANCE” A. Einstein

EPREPR

AA BB

A B

b b’a a’COINCIDENCE

CCOINCIDENCE

C

F11-3

F5-3

PBSR()

D

D

H

V

+

-

DICOTOMIC MEASUREMENTON A SINGLE PHOTON:Click (+) : a = +1Click (-) : a = -1

F4-3

+1 -1

OPTICAL STERN - GERLACH

L(/n)

SINGLE SPIN:

Riemann scalar curvature of the Top:

Lagrangian:

Metric Tensor:

Generalized coordinates: :

(Euler angles)

Euler angles for a rotating body in space

TWO IDENTICAL SPINS

Metric Tensor:

EPR 2-SPIN STATE

(Singlet : invariant under spatial rotations).

=

=

WEYL’s POTENTIAL CONNECTING TWO DISTANT SPINS (in entangled state):

Spatial (x,y,z) terms: (non entangled)

Euler – angles term (ENTANGLED !)

WEYL’s CURVATURE ASSOCIATED TO THE EPR STATE:

Rw =

State of two spinning particles acted upon by Stern – Gerlach (SG) apparatus # 1

Changed

Changed

Changed by SG into:

Under detection of, say , the Weyl curvature acts

nonlocally on apparatus # 2 and determines the EPR correlation !

“Zitterbewegung” (Trembling motion *)

* ) E. Schrödinger, Sitzber. Preuss. Akad. Wiss Physik-Math 24, 418 (1930)

6Wm R

c a c

6. .

( . , . .53,157 (1929).

ha Compton w l

mc mc

Klein paradox on e localization

O Klein Z Phys

Oscillation frequency:

2

2 2 2

: ;

10 0

2

:

1( , ) ( )

2

i jW ij

ijW Wi j

i i j i jW ij i W iji

ji W ij ii

HAMILTON JACOBI EQUATION

Lagrangian L R g q q L L

S SL d L d R g R

q q

Find the Hamiltonian Function

LH p q q L q R g q A L R q q g

q

Lp R g q A

q

1 ( )j ijW j jq R g p A

2 1

:

1 1( )( ) ( )( )

2 2( , )

( ) :

( ); : ( )

[

ik jl klW ij W k K l l W k K l l

i i i i W

i j ii i i

ij

iij

Hamiltonian

H R g R g p A p A g R g p A p A

p S H S q R

S q Solution of the Hamilton Jacobi Equation

q q qq f q Choose as parameter ds g

s s

qg

s

1

2( )] / [ ( )( )] .(8)mnj m nj m n

S e S e S eA g A A Eq

c c cq q q

ELECTROMAGNETIC LAGRANGIAN

where the particle’s mass is replaced by the Weyl’s scalar curvature:

In place of L assume a conformally - invariant Lagrangian:

gqg i )(

DO EXTEND THEORY TO 2 - PARTICLE ASSEMBLY !

2 – PARTICLE METRIC TENSOR

2 – PARTICLE LAGRANGIAN

2 – PARTICLE LAGRANGIAN

2 - PARTICLE BELTRAMI - DE – RHAM - KLEIN – GORDON EQUATIONS

“Zitterbewegung” (Trembling motion *)

* ) E. Schrödinger, Sitzber. Preuss. Akad. Wiss Physik-Math 24, 418 (1930)

6Wm R

c a c

6. .

( . , . .53,157 (1929).

ha Compton w l

mc mc

Klein paradox on e localization

O Klein Z Phys

Oscillation frequency:

THE NEW YORKER COLLECTION Ch..Addams 1940

effeffceff

III

DkkDNbkDnsdk

BAI

)1(

cos2

000

2 1

:

1 1( )( ) ( )( )

2 2( , )

( ) :

( ); : ( )

[

ik jl klW ij W k K l l W k K l l

i i i i W

i j ii i i

ij

iij

Hamiltonian

H R g R g p A p A g R g p A p A

p S H S q R

S q Solution of the Hamilton Jacobi Equation

q q qq f q Choose as parameter ds g

s s

qg

s

1

2( )] / [ ( )( )] .(8)mnj m nj m n

S e S e S eA g A A Eq

c c cq q q

Weyl’s affine connection and curvature:

:dim10

''

ensionsnDincalculatedbecanRcurvature

scalarsWeyloveralltheconnectionaffinesWeylBy

W

ijk