Entanglement
Disentangled
by Spacetime
Vortices
An exploration by John Carroll, Cambridge University
Engineering Department, Cambridge CB2 1PZ, UK
© jec2001
Motivation
Engineering of quantum computers
needs 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 ??
© jec2001
Heuristics of space-time vortices
t = 0; z = 0
q
Now
Future
t = T/3; z = L/3
Past Now Future
t = T; z = L
t = 2T/3; z =2L/3
Past
Now
Past
q
Now
‘now’ fields add; ‘future’/ ‘past’ fields cancel
Polarisation ‘q’ known right around vortices
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-space
gives classic Maxwell; similar algebra with
3d-time + 3d space gives Modified Maxwell
E=
E1x E2x E3x
E1y E2y E3y
E1z E2z E3z
Spatial vectors attached
to each temporal direction
E1x E1y E1z
Etr= E2x E2y E2z
E3x E3y E3z
Temporal vectors attached
to each spatial direction
Modified Maxwell
for 3d time + 3d space
curlspace E = [curltime Btr ]tr
curlspace B = [curltime Etr ]tr
B is ‘3t+3s’ pseudo-‘vector’ counterpart of E.
curltime counterpart of curlspace
© jec2001
‘Modified Maxwell’
Any single field component F:
(t12 + t22 + t32 ) F
= (x2 + y2 + x2 )F
Set Ot3 = Ot ; (t12 + t22) = mo2
obtain Klein Gordon Equation
w2 = mo 2 + k.k : E2 = mo2 + p2
classic relativity! (c=1 =  units)
Recovered ‘Maxwell’
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 ).
2. No direct experimental evidence.
Objections to 3d time
Proposal : No classical
1. Temporal rotations could violate measurements can distinguish
energy conservation: such rotations orientation in transverse time:
inhibited: need excess energy.
Hence can only measure terms like
(Eric Cole – Leeds University)
(Eclassiceiq) * . (Eclassiceiq) =
Proposal : preferred collective
Eclassic* .Eclassic
temporal axis: Ot3 Ot
q : rotation in transverse time. © jec2001
Normal Modes of ‘Recovered Maxwell’
Follow Cohen-Tannoudji et al “Photons and Atoms” CNRS ’87 /Wiley ’89
single k-vector, forward normal modes:
a(k,t) = a(k) exp[i(k.r – kt)] ; b(-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
a(k,t), b(-k,t) are now independent analytic complex vector fields.
Poynting’s Theorem & Modes
Energy density U averaging over volume V denoted by < >
U(k) = < a(k, t)*. a(k, t) + b(–k, t)*. b(–k, t) >
Average energy transfer P (Poynting vector in k direction)
P(k) = < a(k, t)*. a(k, t) – b(–k, t)*. b(–k, t) > (classically zero)
U and P invariant to orientiation of a and b in transverse time
Symmetry requires U(-k)=U(k) : +/- k solutions inseparable
Causality appears to be violated!
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Mode Promotion/Demotion/Annihilation
Select a vector k : define a† = A exp[i(k.r – kt)] ; a = B exp[– i(k.r – kt)]
a† a(k0) = a(k0+k) k0||k : promotes k-vector & frequency) by k & k
a a(k0) = a(k0 – k)
demotes k-vector & frequency) by k & k
provided that always (k0+k) > 0 and (k0 – k) > 0
If (k0 – k) < 0: analytic complex function theory forces aa = 0 
annihilation discovered
‘Normalise’ arbit. const. A and B  STVs promoted from a ground
state have frequencies +/– kN = +/– (k0 + Nk) > 0
(integer N).
Quantisation and Causality
Classic localisation with adjacent frequencies kN and kN +1
Rewrite ‘forward’ waves as a(k)  a(kN+1) and b (–k)  b(–kN)];
Rewrite ‘reverse’ waves as a(–k)  a (–kN–1) and b (k)  b(kN)];
Envelope travels
at group velocity
© jec2001
Set one unit of averaged ‘forward’ energy transfer for P
< a (kN+1) * a(kN+1) - b (-kN) * b (-kN) > = 1
Quantisation and
Causality
continued
=1
Insist P zero for averaged energy transfer in ‘reverse’ direction:
<a (-k-N+1) * a(-k - N+1) - b (kN) * b (kN)>
=0
=0
Eliminate b modes in favour of positive frequency a modes
Average energy UN =
< a(kN+1)*.a(kN+1) + b(–kN)*.b(–kN) > = < a(kN)*[a a† + a† a ]a(kN) >
Average energy transfer
PN = U0 = < a(kN)*[a a† – a† a ]a(kN) >
Postulate Uo = 1 unit : UN =(N+½)U0 ; k0 = ½ k : like Quantum Theory!
© jec2001
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.
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
a* . a = (a + + a–)*.(a + + a–)
= a + * . a + + a - * . a– + mixed terms
Invariance to rotation in transverse time
requires mixed terms a +*.a- = 0
This correlates Stokes parameters.
(S +1 + S +2 + S +3 )
= - (S –1 + S –2 + S – 3)
E– = E'1 – i E'2
tprincipal
E+ = E1 + i E2
tprincipal
t2
time
kz
z
t1
‘positive’
chirality
t2
t1
‘negative’
chirality
kz
By+
y
x E
x+
space
y
By-
Exx
Stokes parameters determine
polarisation in a way that is
invariant to rotation in
transverse time. (See appendix)
Hence spatial polarisation
of +/- chiralities is correlated
Relevant polarisations for net zero spin
© jec2001
Entanglement & Spacetime Vortices
space
incoherent ground state: spacetime vortices ~ (k0)–1 dimensions)
for correlated pair: polarisations not set : freedom of 3d time
STVs: + & – chiralities:
extends over coherence lengths
+/- chiralities carry correlated polarisations: net 0 spin
energy exchange requires interference of kN kN+1 etc STVs
interference propagates at group velocity: ensures causality
‘R’ detected – energy in
one chirality removed
‘L’ detected: energy in
remaining chirality: polarisation
correlated with ‘R’ ; net 0 spin.
t time
NB schematic!
© jec2001
Conclusions
 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’
Acknowledgements
John Baldwin, Cavendish Laboratory Eric Cole, University of Leeds Shaun Ffowcs-Williams Engineering Department
Jeremy 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
Stokes parameters determine polarisation
a x * a x + a x * a x = S 0 a x * a x - a x * a x = S3
ia x * a y - ia y * a x = S 2 a x * a y + a y * a x = S 1
Invariant to rotation in transverse time. True for +/- chirality
*
a+/-x a+/-y a+/-x
= 1+s1S1+/- +s2S2+/- +s3S3+/- = ½ (1 + S+/- .s)
a+/-y
s are Pauli S =
*
*
a
a
+/+x
-x
a+x a+y
a-x a-y
matrices
a+y
a-y
= 1 + S+.S- + (S++S-).s + i(S+x S-) .s
*
a+† a-= a+x
a-x a-y
=0
a+y
implies S+ and S- are anti-parallel (correlated). In interpreting this,
remember that chirality has changed in space as well as in time.
S1+/S2+/S3+/-
© jec2001
Appendix:Two Slit Interference
‘Poynting’ vector now has two real components E1 x B2 – E2x B1
If {E1 B2} symmetric
{E2 B1} asymmetric
screen
(a
{-E2L),-B1L}
forward + reverse fields
{E1R ; B2R} {E2R ; B1R}
{E2R, B1R}
cancel on right.
{E1L ; B2L} {E2L ; B1L}
add on left.
{E1L, B2L}
{E1R, B2R}
source
‘minimum unit’ of
E1x B2 forward energy
detected on screen
Interference patterns as
normal provided that the
fields from each slit reach
the screen.
screen
(b
)
energy transfer could
pass entirely through
left hand slit.
{E1L, B2L}
{E1R, B2R}
source
© jec2001
Selected References
Truesdell C ‘The Kinematics of Vorticity’ Indiana Press, Bloomington 1954 p58
Weinberg, N.N. ‘On some generalisations of the Lorentz Transformations’ Phys.Lett. 80A 102-104
Strnad, 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 1982
Cole E.A.B. ‘Generation of New Electromagnetic Fields in Six Dimensional special relativity’ Il Nuovo Cimento vol 95 1985 p105–117
Cole E.A.B. 1980 ‘New Electromagnetic Fields in Six–dimensional Special Relativity’ Il Nuovo Cimento 60 1–12
Boyling J.B, Cole E.A.B ‘6-Dimensional Dirac-Equation’ International Journal Of Theoretical Physics 32: (5) 801-812 May 1993
Patty 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 17 157-180
Cohen-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