Volume 114A, numbe~ 6 PHYSICS LETTERS 3 March 1986

SUPERFLUID V A C U U M CARRYING REAL E I N S T E I N - D E BROGLIE WAVES

K.P S I N H A a E.C.G. S U D A R S H A N a,h and J.P. V I G I E R c

'~ Indian Institute of Sciences, Bangalore, India b Matscience, Madras, India " Institute Henri Poincarb, 11, rue Pierre et Marie Curie, Paris, France

Received 6 December 1985; accepted for publication 10 December 1985

An earlier superfluid aether model pairing fermionic and antifermionic fields is invoked to explain Rauch's time dependent neutron interference results which now suggest that microobjects are waves and particles simultaneously. The covariant superfluid provides a medium which carries real Einstein-de Broglie "pilot" waves. Further consequences are discussed.

Recent experiments now strongly suggest [1] (if one accepts energy momentum conservation in indi- vidual microphenomena) that (contrary to one o f the basic concepts of the Copenhagen interpretation) neutrons appearing one by one in a double slit inter- ference experiment really behave like particles which travel through one slit only ... so that their subsequent distribution on an interference pattern can only be explained by the existence o f a real physical surround- ing ff field (called "gespenster" field or "pilot wave" by Einstein [2] and de Broglie [3] passing through both slits simultaneously. This ~b field

(1) is associated with a nonlocal "quantum poten- tial" distribution which modifies their time like trajec- tories in EPR type situations [8], and

(2) can and should be described by real collective motions propagating on a subquantal chaos, or aether, which corresponds to a realistic reinterpretation of the quantum "vacuum states".

The minimum physical constraints on any such realistic vacuum model are clear. Such models:

(1) Should be invariant under the full causality group G = T ® L t ® D + P, where T denotes transla- tion, L t the orthochronous Lorentz group, D the dila- tation and P the space symmetry.

(2) Should not manifest any observable quantum number. Namely spin, isospin, colour, charge, etc. Moreover, to fit observations:

(3) The vacuum should contain all possible qua rk -

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lepton states if we accept the premise that bosons are built out of fermion-antifermion pairs.

(4) The vacuum has quark- lepton symmetry which is not broken in the ground state.

(5) The vacuum contains the correlations o f ordered states (i.e. Quantum potential many body action-at- a-distance correlations) and hence provides the super- luminal correlations now experimentally needed to interpret Aspect's [4] experiments on Einstein, Rosen and Podolsky's paradox.

This superfluid presents no friction to macroscopic objects. Any small viscosity would be related to the presence o f the normal fluid component. It would only show up at the cosmological scale.

In order to answer the requirement o f point (1) and (2), we use the superfluid aether model, with a small rest mass m 0 ~ 0 o f the basic fermion and antifermion fields [5,6]. We take a constant surface distribution of four-momentum of the fermion (antifermion) mass shells and constant surface distribution on the spin shell at each space-t ime point. (See fig. 1 o f ref. [7,8]). Accordingly the harniltonian has the following form:

H = ~_J e k C ~ - ck~__ + ~ ekd~a+dkcr ÷

-- ~ V ( k , k ' ) c ~ ' , o _ d q _ k ' , o + d q _ k , a + C k , o _

+ Av' /cX,

0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume l14A, number 6 PHYSICS LETTERS 3 March 1986

where the symbol for summation represents constant surface density over the mass shell and the spin shell [7]. As before ( c~a , cko ), (d~o +, dka+) represent fermion and antifermion (-creation, annihilation) oper- ators respectively with momentum k and spin o_ (for ferrnion) and o+ (for antifermion) and single par- ticle energies e~) = e~ d) = ek;V(k, k') is the funda- mental interaction term [6]. The AvrZ-g/cK takes care of the rest mass and the scalar particle density: A rep- resenting the well-known "cosmological constant" of Einstein which can be considered constant in the lab- oratory frame, x/-L-'g-denoting the square root of the determinant of the metric and cK the gravitational constant.

To meet requirements (3), (4), (5) we can assume, for example that these vacuum particles are related to stochastic harmonic oscillators following Feynman's suggestion. As shown by Cufaro-Petroni et al. [9] this leads to a fermionic state containing 8 quark (+ anti- quarks) and 8 leptons (+ antileptons) which reproduce the Pati-Salam integer charge classification [9]. For the last requirements, we enclose these particles in rigid spherical shells [8], which allow causal action-at- a-distance and the superluminal correlations estab- lished by Aspect's experiments [4].

For a relativistic treatment of this sea of fermions and antifermions, we can follow the method de- scribed by Bailin and Love [10], with the free fermion spinor field ~ described by the lagrangian

zf~ee = ~ ( i 3 ' % - m 0 ) ~ + u~3'0 ~,

where m 0 is the bare rest mass for fermions and anti- fermions,/a the chemical potential and the 3' are Dirac matrices. Ignoring internal degrees of freedom (e.g. isospin, colour etc.), we consider a singlet pair of ferrnion-antifermion in JP = 0 + state. The general gap matrix, for attractive interaction between particles/ antiparticles, will have the form

A0+(n) = A1 3'5 + A2 n *r 7075 + A3 7075,

which depends only on the direction n (unit vector) and not on the magnitude of wave vector. The gap, for some simplified model, turns out to be at T = 0

I A0+IT=0 = 2% exp{-½ [(dn/de)ac 1 ] -1} ,

where e 0 is the energy cut off similar to BCS theory and dn/de is the density of states and ac 1 is a com- plicated expression giving the effective interaction

[ 10,11 ]. In writing this we have chosen a model-in- dependent form such that A °+ = A 1 - (PF//a) A 2 - (mo/la) A3, PF being the Fermi momentum.

The corresponding critical temperature T c for I A°+I -+ O, i.e.

T C = 1.14 (eo/kB) exp {-- [ ½ (dn/de)ac I ]-1 }

is similar to the BCS-line expression for the nonrela- tivistic situation.

It is tempting to identify these pairs of fermions[ antifermions with the Higgs particles in the background aether. In fact, a connection with the Higgs field and the old-fashioned aether in the context of the Salam- Weinberg theory has recently been made [12]. Let us now consider various types of quasiparticles and col- lective excitations (pilot waves, particle like solitons, etc.) over the ground state of the vacuum. Since vari- ous Bose-like excitations with JP equal to 0 - , 1- , 2 + etc. have already been discussed elsewhere [6] we limit ourselves here to single fermion waves. The break- ing of a quark-lepton pair in the JP = 0 + state, will cost 2 [ A 0+ [ = 2mc 2, where m is the effective mass of the fermion pair elements (bare mass m 0 included). Now the pairing is in the momentum state and the fermion and antifermions have opposite momentum and spins. Accordingly, when the pairs break each par- ticle is not created (manifested) at the same space- time point. The breaking process can be considered as a consequence of vacuum fluctuations at a nonzero temperature. In this connection the reference to the work of Baracca and Bohm is relevent [13]. In such a process after some time due to vacuum fluctuations the pair function

¢~a(x 1 ,x 2) - , ~0~ (x 1 ) ea(x 2 )

breaks into independent fermions. These fluctuations give rise to a stochastic background over the superfluid ground state (quantum aether). Clearly the above model of the quantum aether also yields a justification of Einstein's assumption that no particle can exceed the velocity of light. The critical velocity of a super- fluid can indeed be estimated from ref. [5] to be

o C ~lAI/~tk,

where/ik is the momentum of excitation. For k of the order of inverse Compton length we have o C = mc2/mc = c which is just the speed of light. Above this critical velocity superfluidity is destroyed ... so

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Volume 114A, number 6 PHYSICS LETTERS 3 March 1986

that no particle excitat ion can supersede the velocity o f light•

When we consider the superfluid along with fluc- tuations and excitations over the ground state for T

0, we have a two fluid model with a fraction o f the normal component in the universe (manifest observ- able matter)• This situation will give rise to possible interesting cosmological effects such as non-Doppler red-shifts resulting from a small friction of the vacuum• This would change the velocity o f a photon with dis- tance since photon velocity would depend on frequen- cy (even for a free photon) if the photon rest mass, m~, were different from zero as results from the E ins te in -de Broglie relation

hv = m3'c2

x/1 _ , 2 /c 2 ~ph j

The model opens new questions which should be an- swered. A first set of questions is connected with the wave/particle duality. The particle aspect will require in de Broglie's views the description of a soliton beat- ing in phase with the surrounding wave in the super- fluid. For this we will need a nonlinear Dirac equation, which has soliton-like solutions that satisfy the super- posit ion principle. The next set of problems is to con- sider a gauge theoretical model for the vacuum state and explore the possibility o f deducing a mass spec- t rum for quarks and leptons.

Theoretical Studies, Indian Institute of Science, Bangalore for the invitation which make this collabora- t ion work possible•

References

[1] G. Badurek, H. Rauch and J. Summhammer, Phys. Rev. Lett. 51 (1983) 1015; C. Dewdney, Ph. Gudret, A. Kyprianidis and J.P. Vigier, Phys. Lett. 102A (1984) 291; Institut Laue-Langevin, Grenoble, unpublished.

[2] A. Einstein, The Born-Einstein letters (Macmillan, London, 1971) p. 158.

[3] L. de Broglie, Non-linear wave mechanics (Elsevier, Amsterdam, 1960).

[4] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47 (1981) 460;49 (1982) 91.

[5] K.P. Sinha, C. Sivaram and E.C.G. Sudarshan, Found. Phys. 6 (1976) 65,717.

[6] K.P. Sinha and E.C.G. Sudarshan, Found• Phys. 8 (1978) 823.

[7] J.P. Vigier, Lett. Nuovo Cimento 29 (1980) 467. [8] N. Cufaro Petroni and J.P. Vigier, Quantum, space and

time - the quest continues, eds. A.O. Barut et al. (Cambridge Univ. Press, Cambridge, 1983) p. 509.

[9] N. Cufaro Petroni, Z. Maric, D. Zivanovic and J.P. Vigier, J. Phys. A14 (1981) 501.

[10] D. Bailin and A. Love, Phys. Rep. 107 (1984) 325. [ 11 ] K.P. Sinha, Proc. Int. Symp. on Theoretical physics,

Bangalore (1984), to be published. [12] K. Moriyasu, An elementary primer for gauge theory

(World Scientific, Singapore, 1983) p. 120. [13] A. Baracca, D.J. Bohm, B.J. Hiley and A.E.G. Stuart,

Nuovo Cimento 28B (1975) 453.

One o f us (JPV) would like to thank the Centre for

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