Entanglement Disentangled by Spacetime VorticesAn exploration by John Carroll, Cambridge UniversityEngineering Department, Cambridge CB2 1PZ, UK

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t = 0; z = 0 t = T/3; z = L/3 t = 2T/3; z =2L/3 t = T; z = L

Past Now Future Past Now Now Future Past Now

‘now’ fields add; ‘future’/ ‘past’ fields cancel Polarisation ‘’ known right around vortices

Heuristics of space-time vortices

Engineering of quantum computersneeds an understanding of how entanglement can give instantaneous communication between photons about their state of polarisation. Could there be a circulation of fields in time ??

Motivation

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Circulation in time implies vorticity in time.Vorticity requires 3 dimensions [curl (fields)] Hence need to explore 3d time. Geometric (David Hestenes) Algebra for 1d-time+3d-spacegives classic Maxwell; similar algebra with 3d-time + 3d space gives Modified Maxwell

Spatial vectors attached to each temporal direction

E=

Temporal vectors attached to each spatial direction

Etr=

Modified Maxwell for 3d time + 3d space

B is ‘3t+3s’ pseudo-‘vector’ counterpart of E. curltime

counterpart of curlspace

curlspace E = [curltime Btr ]tr

curlspace B = [curltime Etr ]tr

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E1x E2x E3x

E1y E2y E3y

E1z E2z E3z

E1x E1y E1z

E2x E2y E2z

E3x E3y E3z

‘Modified Maxwell’Any single field component F: (t1

2 + t22 + t3

2 ) F = (x2 + y2 + x2 )F Set Ot3 = Ot ; (t1

2 + t22) = mo

2 obtain Klein Gordon Equation = mo 2 + k.k : E2 = mo

2 + p2

classic relativity! (c=1 = units)

Recovered ‘Maxwell’

2. No direct experimental evidence.Proposal : No classical measurements can distinguish orientation in transverse time: Hence can only measure terms like

(Eclassicei * . (EclassiceiEclassic* .Eclassic

: rotation in transverse time.

Objections to 3d time1. Temporal rotations could violate energy conservation: such rotationsinhibited: need excess energy.(Eric Cole – Leeds University)Proposal : preferred collective temporal axis: Ot3 Ot

rest mass = 0: t1 = t2 = 0:E3=0=B3. Real spatial vectors E1 and E2

associated transverse times Ot1/Ot2.Form complex vectors: Eclassic= E1+i E2; Bclassic= – i B1 + B2

curl(Eclassic) = – t(Bclassic); curl(Bclassic) = t(Eclassic ).

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Normal Modes of ‘Recovered Maxwell’Follow Cohen-Tannoudji et al “Photons and Atoms” CNRS ’87 /Wiley ’89

single k-vector, forward normal modes:(k,t) = a(k) exp[i(k.r – kt)] ; (k,t) = b(k) exp[i(k.r + kt)]

k > 0 ; t = t3 ; i rotation through 90o in transverse time;

Eclassic , Bclassic now in general complex so that

(k,t), (k,t) are now independent analytic complex vector fields.

Poynting’s Theorem & Modes Energy density U averaging over volume V denoted by < > U(k) = < (k, t)*. (k, t) + (–k, t)*. (–k, t) > Average energy transfer P (Poynting vector in k direction) P(k) = < (k, t)*. (k, t) – (–k, t)*. (–k, t) > (classically zero)

U and P invariant to orientiation of and in transverse time Symmetry requires U(k)=U(k) : +/ k solutions inseparable Causality appears to be violated! © jec2001

Mode Promotion/Demotion/Annihilation Select a vector : define a† = A exp[i(.r – t)] ; a = B exp[– i(.r – t)] a† (k0) = (k0+) k0|| : promotes k-vector & frequency) by & a (k0) = (k0 – ) demotes k-vector & frequency) by &

provided that always (k0+) > 0 and (k0 – ) > 0If (k0 – ) < 0: analytic complex function theory forces a = 0

annihilation discovered‘Normalise’ arbit. const. A and B STVs promoted from a ground state have frequencies +/– kN = +/– (k0 + N) > 0 (integer N).

Quantisation and CausalityClassic localisation with adjacent frequencies kN

and kN +1

Rewrite ‘forward’ waves as (k)(kN+1) and (–k)(–kN)];

Rewrite ‘reverse’ waves as (–k)(–kN–1) and (k)(kN)]; Envelope travelsat group velocity

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Quantisation and Causality continued

Set one unit of averaged ‘forward’ energy transfer for P<(kN+1) * (kN+1) (kN) * (kN) > = 1

Insist P zero for averaged energy transfer in ‘reverse’ direction: <(kN+1) * (k N+1) (kN) * (kN)> = 0

Eliminate modes in favour of positive frequency modes

Average energy UN =

< (kN+1)*.(kN+1) + (–kN)*.(–kN) > = < (kN)*[a a† + a† a ](kN) >

Average energy transfer PN = U0 = < (kN)*[a a† – a† a ](kN) >

Postulate Uo = 1 unit : UN =(N+½)U0 ; k0 = ½ like Quantum Theory!

Uni-directional energy flow forces standard formalisms of quantum theory Energy transfer requires vortex interference:(kN, kN+1)Interference travels at group velocity restores causality.

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Chirality Solutions to modified Maxwell permit two independent chiralities ±.Analysis to-date applies to both.Chirality in space and time tied together. Both ± exist side by side.Hence now must write* . = ( + + –)*.( + + –)

= + * . + + * . – + mixed termsInvariance to rotation in transverse timerequires mixed terms +*.= 0 This correlates Stokes parameters.(S +1 + S +2 + S +3 )

= (S –1 + S –2 + S – 3)

Hence spatial polarisation of +/- chiralities is correlated

Stokes parameters determine polarisation in a way that is invariant to rotation in transverse time. (See appendix)

‘positive’ chirality

t1

t2

tprincipal

E+ = E1 + i E2

t1t2

E– = E'1 – i E'2tprincipal

‘negative’ chirality

time

space

xyEx-

By-

kz

x

y

z

Ex+

By+

kz

Relevant polarisations for net zero spin© jec2001

t time

space

NB schematic!

incoherent ground state: spacetime vortices ~ (k–1 dimensions)

for correlated pair: polarisations not set : freedom of 3d time

‘R’ detected – energy inone chirality removed

‘L’ detected: energy in remaining chirality: polarisation correlated with ‘R’ ; net 0 spin.

energy exchange requires interference of kN kN+1 etc STVs

STVs: + & – chiralities: extends over coherence lengths

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interference propagates at group velocity: ensures causality

+/- chiralities carry correlated polarisations: net 0 spin

Geometric Algebra +3d time: balances temporal/spatial vorticity. Concept of spacetime vortex (STV): spacetime energy circulation Modified ‘Maxwell’ equations permit massive particles (not explored) Massless system recovers almost classical Maxwell Poynting vector now has coupled energy flowing in +/ time Unidirectional energy flow at a measurement forces quantization & causality. Quantum rules discovered not postulated. Transverse time allows two independent chiralities: extra freedom. Entangled photons do not have both polarizations determined

until measured but chiralities are correlated. Measurement of one photon (‘R’) removes one temporal chirality of STVs leaving energy in correlated temporal chirality: gives ‘communication’ between ‘L’& ‘R’

Conclusions

AcknowledgementsJohn Baldwin, Cavendish Laboratory Eric Cole, University of Leeds Shaun Ffowcs-Williams Engineering DepartmentJeremy Carroll , Hewlett Packard for listening and helpful comments.

Anthony Lasenby, Cavendish Laboratory Chris Doran, DAMPT http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/Joan Lasenby, Engineering Department for notes on Geometric Algebra © jec2001

Appendix: Stokes parameters

= 1 + SS+ (S+S + i(SxS

implies Sand S are anti-parallel (correlated). In interpreting this, remember that chirality has changed in space as well as in time.

Stokes parameters determine polarisation x * x x * x = S 0 x * x x * x = S3

i x * y i y * x = S 2 x * y y * x = S 1Invariant to rotation in transverse time. True for +/ chirality

x yx

y

*= 1+S1+/- +S2+/- +S3+/- ½ (1 + S

x

y

* x yx yx

y

*

† x

y

* x y

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= 0

S1+/

S2+/

S3+/

S+/=are Pauli matrices

If {E1 B2} symmetric {E2 B1} asymmetric

forward + reverse fields {E1R ; B2R} {E2R ; B1R}cancel on right.

{E1L ; B2L} {E2L

; B1L} add on left.

‘minimum unit’ of energy transfer could pass entirely through left hand slit.

Appendix:Two Slit Interference

E1x B2 forward energy detected on screenInterference patterns as normal provided that the fields from each slit reach the screen.

‘Poynting’ vector now has two real components E1 x B2 – E2x B1

{-E2L,-B1L}

screen

source

(a) {E2R, B1R}

{E1R, B2R} {E1L, B2L}

screen

source

(b)

{E1R, B2R} {E1L, B2L} © jec2001

Selected ReferencesTruesdell C ‘The Kinematics of Vorticity’ Indiana Press, Bloomington 1954 p58 Weinberg, N.N. ‘On some generalisations of the Lorentz Transformations’ Phys.Lett. 80A 102-104Strnad, J., ‘Experimental-Evidence Against A 3-Dimensional Time’ Physics Letters A, 1983, Vol.96, No.5, Pp.231-232 Cole E.A.B., Buchanan, S.A Space-Time Transformations In 6-Dimensional Special Relativity Jnl Of Phys A- Mathematical And General 15: (6) L255-L257 1982Cole E.A.B. ‘Generation of New Electromagnetic Fields in Six Dimensional special relativity’ Il Nuovo Cimento vol 95 1985 p105–117Cole E.A.B. 1980 ‘New Electromagnetic Fields in Six–dimensional Special Relativity’ Il Nuovo Cimento 60 1–12Boyling J.B, Cole E.A.B ‘6-Dimensional Dirac-Equation’ International Journal Of Theoretical Physics 32: (5) 801-812 May 1993Patty C.E., Smalley L.L., ‘Dirac-Equation In A 6-Dimensional Spacetime - Temporal Polarization For Subluminal Interactions Phys Review D 32: (4) 891-897 1985

Einstein A Podolsky B and Rosen W. ‘Can Quantum mechanical description of physical reality be considered complete’ Phys Rev 47 777-780 Clauser_J.F , Horne M.A. ‘Experimental consequences of objective local theories’. 1974 Vol.10 P.526-535, Physical Rev D Aspect, A., Dalibard, J., Roger, G., ‘Experimental Test Of Bell Inequalities Using Time-Varying Analyzers’ Physical Review Letters, 1982, Vol.49, No.25, pp.1804-1807 Greenberger_DM, Horne_M, Zeilinger_A, ‘Similarities and differences between two-particle and three- particle interference’ : Fortschritte Der Physik-Progress Of Physics, 2000, 48, pp.243-252 Wheeler J.A and Feynman R.P. Interaction With The Absorber As The Mechanism Of Radiation Reviews Of Modern Physics 1945 17 157-180Cohen-Tannoudji, C. Dupont-Roc J. and Grynberg, G Photons and Atoms J.Wiley New York 1989 (originally in French Photons et Atomes 1987 Inter-editions et Editions du CNRS)Cramer, J.G. 1986 The Transactional Interpretation Of Quantum Mechanics, Rev. Mod. Phys. 58, 647– 687.

Hestenes, D. 1985 New Foundations for Classical Mechanics Dordrecht Reidel

Hestenes, D. 1966 Spacetime algebra New York Gordon and Breach

Hestenes, D. 1985 Quantum Mechanics from self interaction Foundations of Physics 15 63-87

Lasenby, A., Doran, C. and Gull, S. Gravity, gauge theories and geometric algebra Phil Trans. R.Soc. Lond. A (1998), 356 , 487-582

Gull, S. Lasenby A. & Doran,C. 1993 Imaginary Numbers Are Not Real – The Geometric Algebra Of Space-Time Foundations Of Physics 25, 1175-1201.

Lasenby A.N Doran C.J lecture notes 2000-2001 http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/

Carroll Spacetime vortices: see http://www2.eng.cam.ac.uk/~jec/spacetimevortices.pdf

http://www2.eng.cam.ac.uk/~jec/spacetimevortices2.pdf © jec2001