Quantum Mechanics and Geometric Interpretations in Weyl Space
The discussion explores Weyl quantum mechanics, Bohm's interpretation of quantum potential, and geometric formulations in Euclidean-Weyl space. It delves into the implications of nonlocal quantum potentials and the nature of metric spaces in shaping quantum phenomena. Concluding with alternative viewpoints on the deterministic nature of quantum mechanics in Euclidean and non-Euclidean spaces.
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Weyl Quantum Mechanics: spin 1/2, EPR and spin statistics from geometry differential geometry and Enrico Santamato Francesco De Martini Napoli, 29/04/2019
Part I Introduction Napoli, 29/04/2019
Bohms interpretation of QM 1 m = 2 + ( ) r i V 2 S t Bohm s Quantum Potential i 2 = = e S = continuity e q uation t m Where this nonlocal potential comes from? 1 m S t ( ) 2 = + + ( , ) r ( , r ) H-J equation S V t Q t B 2 2 2 2 = ( , ) r Q t 2 2 2 m Napoli, 29/04/2019
Bohms Quantum Potential comes from Geometry Schr dinger equation assumes Euclidean space Euclidean space has o Euclidean metric for vector length o No change in vector length in a parallel transport It is in the nature of a metric space to have this parallel transport law Give up this! H. Weyl, Space,Time,Matter, Dover, 1952 Assume instead H. Weyl (1919) = = 2 ( ) = r r a a 2 d d Napoli, 29/04/2019
Bohms Quantum Potential comes from Geometry If = 2 dr in a parallel transport the 3D space with Euclidean metric acquires a nonzero Weyl scalar curvature R = 4 2 W Assume to be a gradient Bohm s Quantum potential = = (ln ) 2 2 = = = 2 8 R Q W B 2 4 4 m Napoli, 29/04/2019
Geometric formulation of QM in Euclidean-Weyl space S = t m Weyl curvature 1 m S t ( ) 2 = + + 2 S V R W 2 Now has a geometrical meaning: it fixes the parallel transport law of vectors in the 3D physical space. Napoli, 29/04/2019
Conclusions (part I) Alternatives 3D physical space is Euclidean. QM is not deterministic. has a statistical interpretation (standard). 3D physical space is Euclidean. QM is deterministic but with nonlocal quantum potential. is like a fluid density. Believe in Weyl s statement It is in the nature of any metric space to have a not trivial parallel transport H. Weyl, Space,Time,Matter, Dover, 1952 3D physical space has Euclidean metric only. QM is deterministic and local. has a geometric interpretation. Napoli, 29/04/2019
Part II The Conformal Quantum Geometrodynamics (CQG) Napoli, 29/04/2019
Weyls geometry (N-dimensions) Two fundamental forms A quadratic form (metric) 2 ds = i j ( ) g q dq dq ij A linear form (parallel transport) = = ( ) q d = i i j ( ) ( ) q ( ) d q g q a a q i ij Napoli, 29/04/2019
Weyls geometry (N-dimensions) Weyl s connections i jk ( ) = + + = i i j k i i i i kj g jk k j jk jk Weyl s scalar curvature ( ) 2 = + ij ij ( 1 ) ( 2 ) R R n g g n g W R i j i j g The Weyl space is curved even if the metric is flat (in Riemann sense, i.e. RR= 0) H. Weyl, Space,Time,Matter, Dover, 1952 It is in the nature of a metric space to be furnished with these affine connections
Weyls geometry (N-dimensions) Assume the Weyl connection integrable 1 i N 1 = = (ln ) i i 2 2 N k k 1 2 2 N N = = + 2 k k R R R Q W R R B 2 Weyl Rieman kcovariant derivative w.r.t. gij gijis used to lower/raise the indices Napoli, 29/04/2019
Weyls gauge invariance The Weyl connections are invariant under Weyl s gauge transformations ij g Conformal transformation g ij 1 2 + = + ln i i i i i 2 2 N 2 Tensors T transforms simply under Weyl s gauge if w(T) Weyl s weigth ( ) w T T T Examples: w(gij) = 1, w(RW) = 1, w( ) = (N 2)/2, and i does not transform simply. July, 9 th 2013 A quantum day, Napoli (Italy) 12
Are the CQG equations Weyl invariant? 1 m S t S q S q = + 2 ij g R W j j 2 W = 0 W = 1 No ( ) g 1 m S q = ij g g i j t q W = 0 W = +1 Napoli, 29/04/2019
Relativistic CQG S x S x + + = = diag( 1,1,1,1) 2 2 2 0 g R m c g W 1 S x = 0 g g x g W(m2) = 1 !? 2 2 1 m c Klein-Gordonequation gg = + R R 2 x x g Weyl invariance is hidden Napoli, 29/04/2019
Relativistic CQG Gauge invariant action principle: I = 0 S x S x = + + 2 2 2 4 I g R m c gd x W S x S x + + = Variation w.r.t. 2 2 2 0 g R m c W = 1 S x 0 g g Variation w.r.t. S x g Variation w.r.t. g leads to incompatible field equations unless m = 0 Napoli, 29/04/2019
Extended gravity in (4+K)-D S q S q Gauge invariant action = + 2 i j N I g R gd q W i j = 0 g g ij 0 g Space-like extra coordinates is the metric of a homogeneous space (e.g. a group G) The constant Riemann curvature RR of plays role of mass Napoli, 29/04/2019
Field equations in (4+K)-D + = 2 k 0 S S R Weyl gauge invariance is manifest Weyl curvature k W ( ) = = k k 0 j S k k iS = e Riemann curvature + = = 2 i R i R ( ) ( ) x = = 2 Weyl gauge invariance is hidden k 2 2 m c ( ) = + = 2 2 R k R 2 Klein-Gordon Napoli, 29/04/2019
Conclusions (part II) The relativistic QM is Weyl gauge-invariant Only gauge-invariant quantities have an objective physical meaning, e.g. the phase action S and the current vector density = | |2 is not Weyl-gauge invariant. It has no objective physical meaning. Born s statistical interpretation of = | |2 has no objective physical meaning The current density jiis Weyl invariant and has objective physical meaning (count rate in particle detectors) = i ij j g S g i Napoli, 29/04/2019
Conclusions (part II) De Broglie- Bohm nonlocal pilot wave The CQG is local? Assume space M4 G in (4+K)-D = + 2 2 2 R R Q 2 = = ( , ) r Q t W G B B 2 2 S q S q = + 2 i j N I g R gd q Gauge invariant Klein-Gordon W i j New metric Pass to the gauge where = 1 2 + S q S q ij = + = 2 N I g R gd q g g 2 K Gravitational action of the scalar field S R ij ij i j Napoli, 29/04/2019
Part III Spin 1/2 Napoli, 29/04/2019
Dirac equation Spin degrees of freedom: the six Euler angles of the Lorentz group SO(3,1) Space M4 SO(3,1) in 10-D. Coordinates qi= {x ,y } = {x , } S q e c S q e c = + iS iS 2 2 ij N I g A A R gg d q (3,1) = = i j W SO i j e e = = ( ) y = { ( ), ( x E r H { , )} { , ( ) x ) } )} A A A A A A x i r r e c e c + + 2 2 i j Klein-Gordon (4 )-D N g i A i A R i i j j R E. Santamato and F. De Martini, Found. Phys. 43, 631 (2013) Napoli, 29/04/2019
Dirac equation 1/2 = + (0,1/2) 1 (1/2,0) 1 s p p s p p ( , x y ) ( ( )) ( ) x ( ( )) ( ) D y D y x s L R = 1/2 p 1/2 L ( ) ( ) ( ) ( ) x x x D D + D D = + = 0 D D e c 1/2 L = ( ) x Dirac's spinor = D i A m D 1/2 R x 1/2 2 ( ) ( ) R = 3 4 + + 2 2 m o F F 2 2 2 c E. Santamato and F. De Martini, Found. Phys. 43, 631 (2013) Napoli, 29/04/2019
EPR paradoxes Because the CQG is deterministic and complete the riddle of EPR paradoxes is automatically solved E, Santamato, F. De Martini, Solving the nonlocality riddle by conformal quantum geometrodynamics , J, Phys. Conf. Ser., 442, 012059 (2013) F. De Martini and E.Santamato Interpretation of Quantum Nonlocality by Conformal Quantum Geometrodynamics , Int. J. Th. Ph.,53,3308 (2014) F. De Martini and E.Santamato, Nonlocality, No-Signalling and Bell's Theorem investigated by Weyl's Conformal Differential Geometry , Physica Scripta, T163, 014015 (2014) F. De Martini and E.Santamato, Violation of the Bell Inequalities by Weyl Conformal Quantum GeometrodynamicsA Re-Interpretation of Quantum Nonlocality , J. Adv. Phys., 4, 272 (2015) Napoli, 29/04/2019
Spin statistics Napoli, 29/04/2019
Spin statistics In SO(3,1) use Euler angles y = { , , , , , }. The H-J the continuity equations of the CQG in (4+6)-D do not contain ( is ignorable) = + = 0( ) e 0 S s S q s s = intrinsic helicity is ( , ) x ( ) y z s The helicity s is strictly constant From the harmonic expansion above we see that s = 1/2 for the spin 1/2. For general spin, the harmonic expansion in SO(3,1) yields s integer or half-integer. Napoli, 29/04/2019
Spin statistics Single particle s = = + (0, ) 1 ( ,0) s 1 is s s p p s p p ( ) ( , x y ) ( ( )) ( ) x ( ( )) ( ) x q e D B z D B z s s L R = p s = ( , , , , ) ( ) = SO(3,1 )/SO( 2) z B z Because s > 0, the rotation of must be always counterclockwise! Two identical particles s + = + ( ) is (0, ) 1 (0, ) 1 s s p s s p ( , q q ) ( ( )) ( ( )) ( , ) dotted term s e D B z D B z x x 1 2 1 2 1 2 , 1 2 s p p 1 2 1 2 = , p p s 1 2 Napoli, 29/04/2019
Spin statistics The H-J and continuity equations require = ( S , ) ( , ) q q S q q 1 2 2 1 Because (q1,q2) = (q2,q1) we have also = ( , q q ) ( , ) q q 1 2 2 1 s s Because the rotation of 1 and 2 must be counterclockwise, the exchange of 1 and 2 introduces a factor ( 1)2s in front of s The exchange of the other coordinates introduces no factor Napoli, 29/04/2019
Spin statistics s + ( 1) = + = ( ) is 2 (0, ) 1 (0, ) 1 s s s p s s p ( , ) ( ( )) ( ( )) ( , ) dotted terms q q e D B z D B z x x 1 2 2 1 2 1 , 2 1 s p p 1 2 1 2 = , p p s 1 2 s + ( 1) = + ( ) is 2 (0, ) 1 ( 0, ) 1 s s s p s s p ( ( )) ( ( )) ( , ) dotted t er ms e D B z D B z x x 1 2 2 1 1 , 2 1 p p 2 2 1 p = , p s 1 2 Comparison with s + = + ( ) is (0, ) 1 (0, ) 1 s s p s s p ( , q q ) ( ( )) ( ( )) ( , ) dotted term s e D B z D B z x x 1 2 1 2 1 2 , 1 2 s p p 1 2 1 2 = , p p s 1 2 yields = 2 s ( , ) ( 1 ) ( , ) x x x x , 1 2 , 2 1 p p p p 1 2 2 1 EOP Napoli, 29/04/2019
Perspectives Extend the space to SU(2) U(1) SO(3,1) to include Weinberg s quantum model of leptons Introduce gauge Yang-Mills fields Introduce Higgs field to provide masses Apply Weyl s conformal symmetry to quantum field theory Study gravitational and cosmological implications Napoli, 29/04/2019
Conclusions Weyl s geometry and symmetry shed new light on Quantum Mechanics The statistical Born s interpretation of | |2 has no objective physical meaning and should be replaced by a geometrical interpretation When the CQG is applied to a particle with spin, a new conserved quantity appears: the particle helicity EPR and other QM paradoxes are automatically solved, because the CQG is deterministic and complete The spin-statistic connection can be obtained for any spin exploiting symmetry consideration only Napoli, 29/04/2019
