quantum dynamics and conformal geometry: the “affine quantum mechanics” from dirac’s equation...
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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