Planck Mass Plasma Vacuum Conjecture

F. Winterberg University of Nevada, Reno, Nevada, USA Reprint requests to Prof. F. W.; Fax: (775)784-1398

Z. Naturforsch. 58a, 231 – 267 (2003); received July 22, 2002

As an alternative to string field theories in R10 (or M theory in R11) with a large group and a very large number of possible vacuum states, we propose SU2 as the fundamental group, assuming that nature works like a computer with a binary number system. With SU2 isomorphic to SO3, the rotation group in R3, explains why R3 is the natural space. Planck’s conjecture that the fundamental equations of physics should contain as free parameters only the Planck length, mass and time, requires to replace differentials by rotation – invariant finite difference operators in R3. With SU2 as the fundamental group, there should be negative besides positive Planck masses, and the freedom in the sign of the Planck force permits to construct in a unique way a stable Planck mass plasma composed of equal numbers of positive and negative Planck mass particles, with each Planck length volume in the average occupied by one Planck mass particle, with Planck mass particles of equal sign repelling and those of opposite sign attracting each other by the Planck force over a Planck length. From the thusly constructed Planck mass plasma one can derive quantum mechanics and Lorentz invariance, the latter for small energies compared to the Planck energy. In its lowest state the Planck mass plasma has dilaton and quantized vortex states, with Maxwell’s and Einstein’s field equations derived from the antisymmetric and symmetric modes of a vortex sponge. In addition, the Planck mass plasma has excitonic quasiparticle states obeying Dirac’s equation with a maximum of four such states, and a mass formula of the lowest state in terms of the Planck mass, permitting to compute the value of the finestructure constant at the Planck length, in surprisingly good agreement with the empirical value.

Key words: Planck Scale Physics; Analog Models of General Relativity and Elementary Particles Physics.

1. Introduction: Fundamental Group and 12. Gravitational Self-Shielding of Large Masses 253 Non-Archimedean Analysis 231 13. Black Hole Entropy 254 2. Planck Mass Plasma 233 14. Analogies between General Relativity, 3. Hartree-Fock Approximation 238 Non-Abelian Gauge Theories and Superﬂuid 4. Formation of Vortex Lattice 238 Vortex Dynamics 257 5. Origin of Charge 239 15. Quark-Lepton Symmetries 258 6. Maxwell’s and Einstein’s Equations 239 16. Quantum Mechanical Nonlocality 260 7. Dirac Spinors 243 17. Planck Mass Rotons as Cold Dark Matter 8. Finestructure Constant 248 and Quintessence 265 9. Quantum Mechanics and Lorentz Invariance 249 18. Conclusion 265 10. Gauge Invariance 250 References 267 11. Principle of Equivalence 251

1. Introduction: Fundamental Group and physics, and analogies between condensed matter Non-Archimedean Analysis physics and general relativity [1 – 9]. These analogies suggest that the fundamental group of nature is String (resp. M) theory uniﬁcation attempts are small, with higher groups and Lorentz invariance de- characterized by going to ever larger groups and higher rived from the dynamics at a more fundamental level. dimensions. Very different, but at the present time A likewise dynamic reduction of higher symmetries to much less pursued attempts are guided by analogies an underlying simple symmetry is known from crystal between condensed matter and elementary particle physics, where the large number of symmetries actu-

0932–0784 / 03 / 0400–0231 $ 06.00 c 2003 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com 232 F. Winterberg · Planck Mass Plasma Vacuum Conjecture ally observed are reduced to the spherical symmetry of where l0 > 0 is the finite difference. We can write [11] the Coulomb field. ( + )= hd/dx ( ) We make here the proposition that the fundamental f x h e f x (1.2) group is SU2, and that by Planck’s conjecture the fun- d f (x) h2 h2d2 f (x) damental equations of physics contain as free parame- = f (x)+h + + ··· dx 2! dx2 ters only the Planck length rp, the Planck mass mp and and thus for (1.1) Planck time tp (G Newton’s constant, h Planck’s constant, c the velocity of light): ∆y sinh[(l0/2)d/dx] = f (x). (1.3) ∆x l0/2 hG¯ − r = ≈ 10 33 cm, p c3 We also introduce the average ( f (x + l0/2)+ f (x − l0/2)) hc¯ −5 y¯ = (1.4) mp = ≈ 10 g, 2 G = cosh[(l0/2)d/dx] f (x). hG¯ − t = ≈ 10 44 s. → ∆ /∆ = / = p c5 In the limit l0 0, y x dy dx, andy ¯ y. With d/dx = ∂ we introduce the operators The assumption that SU2 is the fundamental group means nature works like a computer with a binary ∆0 = cosh[(l0/2)∂], ∆1 =(2/l0)sinh[(l0/2)∂] (1.5) number system. As noted by von Weizsäcker [10] with SU2 isomorphic to SO3, the rotation group in R3, whereby then immediately explains why natural space is three- ∆y dimensional. Planck’s conjecture requires that differ- = ∆1 f (x), y¯ = ∆0 f (x) (1.6) ∆x entials must be replaced by rotation invariant finite difference operators, which demands a finitistic non- and 1 Archimedean formulation of the fundamental laws . 2 ∆ And because of the two-valuedness of SU2 there must ∆ = 2 d 0 . 1 ∂ (1.7) be negative besides positive Planck masses2. Planck’s l0 d conjecture further implies that the force between the The operators ∆ and ∆ are solutions of = 0 1 Planck masses must be equal the Planck force Fp 2 4 mpc /rp = c /G acting over a Planck length. The 2 2 d l0 remaining freedom in the sign of the Planck force − ∆(∂)=0. (1.8) ∂2 2 makes it possible to construct in a unique way a sta- d ble “plasma” made up of an equal number of positive The generalization to an arbitrary number of dimen- and negative Planck masses, with each volume in space sions is straightforward. For N dimensions and ∆ = ∆ 0, occupied in the average by one Planck mass. (1.8) has to be replaced by [12] For the finitistic non-Archimedean analysis we pro- N 2 2 ceed as follows: the finite difference quotient in one ∂ l ( ) ∑ − N 0 ∆ N = 0, (1.9) dimension is ∂(∂ )2 0 i=1 i s ∆y f (x + l /2) − f (x − l /2) = 0 0 , (1.1) where ∆x l0 ∆ (N) = . 1 lim 0 1 Non-Archimedean is here meant in the sense of Archimedes’ l0→0 belief that the number π can be obtained by an unlimited progression of polygons inscribed inside a circle, impossible if there is a smallest From a solution of (1.9) one obtains the N dimen- length. sional finite difference operator by 2Since the theory at this most fundamental level is exactly nonrelativistic, the particle number is conserved, outlawing their change 2 (N) ( ) 2 d∆ into other particles. This permits the permanent existence of negative ∆ N = 0 . (1.10) besides positive Planck masses. i l0 d∂i F. Winterberg · Planck Mass Plasma Vacuum Conjecture 233 Introducing N-dimensional polar coordinates, (1.9) be- mass particles of equal sign repelling and those of op- comes posite sign attracting each other3. The particular choice made for the sign of the Planck force is the only one 2 1 d − d l (N) which keeps the Planck mass plasma stable. While ∂N 1 − N 0 ∆ = 0, (1.11) ∂N−1 d∂ d∂ 2 0 Newton’s actio=reactio remains valid for the interaction of equal Planck mass particles, it is violated for the where interaction of a positive with a negative Planck mass particle, even though globally the total linear momen- N tum of the Planck mass plasma is conserved, with the ∂ = ∂2. ∑ i recoil absorbed by the Planck mass plasma as a whole. = i 1 It is the local violation of Newton’s actio=reactio ( ) which leads to quantum mechanics at the most fun- Putting ∂ ≡ x, ∆ N ≡ y, (1.11) takes the form 0 damental level, as can be seen as follows: Under the F = m c2/r 2 +( − ) − ( / )2 2 = Planck force p p p, the velocity fluctuation x y N 1 xy N l0 2 x y 0 (1.12) of a Planck mass particle interacting with a Planck mass particle of opposite sign is ∆v =(Fp/mp)tp = with the general solution [13] 2 (c /rp)(rp/c)=c, and hence the momentum fluc- √ 3−N tuation ∆p = mpc. But since ∆q = rp, and because y = x 2 Z±( − )/ i N(l0/2)x , (1.13) 3 N 2 mprpc = h¯, one obtains Heisenberg’s uncertainty relation ∆p∆q = h¯ for a Planck mass particle. Accord- where Zν is a cylinder function. ingly, the quantum fluctuations are explained by the Of interest for the three dimensional difference interaction with hidden negative masses, with energy operator are solutions where N = 3 and where borrowed from the sea of hidden negative masses. ( ) lim ∆ 3 = 1. One finds that The conjecture that quantum mechanics has its → 0 l0 0 cause in the interaction of positive with hidden nega- √ tive masses is supported by its derivation from a vari- ( ) sinh 3(l /2)∂ ≡ ∆ 3 = √ 0 , ational principle first proposed by Fenyes [14]. Be- D0 0 (1.14) 3(l0/2)∂ cause Fenyes could not give a physical explanation for his variational principle, he was criticized by Heisen- and with (1.10) that berg [15], but the Planck mass plasma hypothesis gives 2 √ a simple explanation through the existence of negative ≡ ∆ (3) = 2 l0 ∂ masses. D1 1 cosh 3 (1.15) l0 2 According to Newtonian mechanics and Planck’s √ conjecture, the interaction of a positive with a nega- sinh 3(l /2)∂ ∂ − √ 0 i . tive Planck mass particle leads to a velocity fluctuation 2 δ˙ = = 3(l0/2)∂ ∂ aptp c, with a displacement of the particle equal to δ =(1/2)a t2 = r /2, where a = F /m . There- ∆ p p p p p p The expressions 1 and D1 will be used to obtain the fore, a Planck mass particle immersed in the Planck dispersion relation for the finitistic Schrödinger equa- mass plasma makes a stochastic quivering motion (Zit- tion in the limit of short wave lengths. terbewegung) with the velocity

2. Planck Mass Plasma vD = −(rpc/2)( n/n), (2.1)

= / 3 The Planck mass plasma conjecture is the assump- where n 1 2rp is the average number density of pos- tion that the vacuum of space is densely filled with itive or negative Planck mass particles. The kinetic en- an equal number of positive and negative Planck mass particles, with each Planck length volume in 3It was shown by Planck in 1911 that there must be a divergent the average occupied by one Planck mass, with the zero point vacuum energy, by Nernst called an aether. It is for this reason that I have also called Planck’s zero point vacuum energy, Planck mass particles interacting with each other by the Planck aether. Calling it a Planck mass plasma instead appears the Planck force over a Planck length, and with Planck possible as well. 234 F. Winterberg · Planck Mass Plasma Vacuum Conjecture ergy of this diffusion process is given by (2.6) and (2.10) is obtained from the Schrödinger equation m m 2 2 2 p 2 = p 2 2 n = h¯ n . ∂ψ 2 vD rpc h¯ 2 2 8 n 8mp n ih¯ = − ψ +Uψ. (2.12) ∂t 2mp (2.2) Repeating the same with the non-Archimedean anal- Putting ∂/∂ → ∆ 2 → 2 ∆ ysis by replacing t 1 and D1, with 1 D h¯ given by (1.5) and 1 by (1.15), leads to the finitistic v = S, (2.3) form of (2.12): mp 2 ∆ ψ = − h¯ 2ψ + ψ. where S is the Hamilton action function and v the ve- ih¯ 1 D1 U (2.13) locity of the Planck mass plasma, the Lagrange density 2mp for the Planck mass plasma is In the limit of short wave lengths one can neglect the potential U. In this limit the dispersion relation ∂S h¯2 h¯2 n 2 L = n h¯ + ( S)2 +U + . (2.4) for (2.13) differs from the on for (2.12), but goes over ∂t 2mp 8mp n into the latter in the limit rp → 0, tp → 0. For a wave field ψ = ψ(x,t) one obtains from (2.13) Variation of (2.4) with regard to S according to 2ih¯ t ∂ ∂ ∂L ∂ ∂L sinh p ψ = (2.14) + = 0 (2.5) t 2 ∂t ∂t ∂S/∂t ∂r ∂S/∂r p 2 √ leads to 8mpc − cosh 3(r /2)∂/∂x r2 p ∂n h¯ p + (n S)=0 (2.6) √ ∂t m sinh 3(r /2)∂/∂x 2 p − √ p 1 ψ, ( / )∂/∂ (∂/∂ )2 or 3 rp 2 x x and for a plane wave ∂n + (nv)=0, (2.7) ∂ ( −ω ) t ψ = Aei kx t (2.15) which is the continuity equation of the Planck mass one obtains the dispersion relation plasma. Variation with regard to n according to ωt 3 ∂L ∂ ∂L sin p = √ (2.16) − = 0 (2.8) 2 ( / ) 2 ∂n ∂r ∂n/∂r 3 rp 2 k √ √ 2 leads to sin 3(rp/2)k · √ − cos 3(rp/2)k . 3(rp/2)k ∂ 2 2 2 2 S + + h¯ ( )2 + h¯ 1 n − n = √ h¯ U S 0 x = (r / )k ∂t 2mp 4mp 2 n n Putting 3 p 2 , this can be written as follows: (2.9) ωt 3 sinx 3sin p = f (x), f (x)= − cosx . or 2 x x √ ∂ 2 2 2 (2.17) S + + h¯ ( )2 + h¯ √ n = . h¯ ∂ U S 0 (2.10) t 2mp 2mp n One has lim f (x)=x and lim f (x)=0, with a maxi- x→0 x→∞ With the Madelung transformation mum of f (x) at x 2.1, where f (x) 1.3. In the limit 2 √ √ x = 0 and tp → 0 one has ω =(crp/2)k , which with iS ∗ −iS 2 ψ = ne , ψ = ne (2.11) mprpc = h¯ is ω = hk¯ /2mp, the dispersion relation in F. Winterberg · Planck Mass Plasma Vacuum Conjecture 235 = . ( ) (| − |) the limit U 0. For x 2 1, where f x has its max- where D q qn is the position eigenfunction with imum, one has k 2.4/rp and hence sin(ωmaxtp/2) the limit 0.57. With the energy operator E = ih¯∆ 1 one then finds lim D(|q − q |)=δ(|q − q |). (2.27) → n n ω rp 0 2¯h maxtp 2 Emax = sin ≈ 1.14mpc . tp 2 By comparison with (2.24) and (2.25) it follows that (| − |) (2.18) with the position eigenfunction D q qn , the position operator is For the transition from classical to quantum me- 1 d chanics we set for the momentum operator in analogy q → q, (2.28) to p =(h¯/i)∂/∂r D1 dq

h¯ which is the same as (2.20). For the potential U p = Di. (2.19) in (2.12) and (2.13), coming from all the Planck mass i particles acting on one Planck mass particle ±mp de- To insure the integrity of the Poisson bracket relation scribed by the wavefuntion ψ±,weset {q, p} = 1 requires then a change in the position op- = ± 2ψ∗ ψ − ψ∗ ψ , erator q to keep unchanged the commutation relation U± 2¯hcrp ± ± ∓ ∓ (2.29) pq− qp = h¯/i, and one has to set which we justify as follows: A dense assembly of pos- ∂ itive and negative Planck mass particles, each of them = i . q r (2.20) occupying the volume r3, has the expectation value D1 p ψ∗ ψ = / 3 2ψ∗ ψ = 2 ± ± 1 2rp, whereby 2¯hcrp ± ± mpc , im- The meaning of the nonlocal position operator (2.20) 2 plying an average potential ±mpc for the positive or can be understood by the position eigenfunctions of negative Planck mass particles within the Planck mass this operator. If ψ (q) is the position eigenfunction and 2 n plasma, and consistent with Fprp = mpc . We thus q the position operator, the eigenvalues q n and eigen- have for both the positive and negative Planck mass ψ ( ) functions n q are determined by particles ψ ( )= ψ ( ). q n q qn n q (2.21) ∂ψ 2 ± = ∓ h¯ 2ψ ± 2ψ∗ ψ − ψ∗ ψ ψ . ih¯ ± 2¯hcrp ± ± ∓ ∓ ± If the position can be precisely measured, one has ∂t 2mp (2.30) ψ ( )=δ(| − |) n q q qn (2.22) To make the transition from the one particle and Schrödinger equation (2.30) to the many particle equation of the Planck mass plasma we replace (2.30) by δ(| − |) = , q q qn dq qn (2.23) ∂ψ 2 ± = ∓ h¯ 2ψ ± 2ψ† ψ − ψ† ψ ψ , ( ≡ / ) ih¯ ± 2¯hcrp ± ± ∓ ∓ ± for which one can write 1 d ∂t 2mp (2.31) dq δ(| − |)= . q q qn qn (2.24) d where ψ †,ψ are field operators satisfying the commu- If q = q one has tation relations n † dq ψ±(r)ψ±(r ) = δ(r − r ), δ(|q − q |)=1. (2.25) d n (2.32) † † [ψ±(r)ψ±(r )] = ψ±(r)ψ±(r ) = 0. For the nonlocal position operator one likewise has 1 Replacing in (2.31) ψ †,ψ by the classical field func- (| − |)= , ∗ D q qn 1 (2.26) tions ϕ ,ϕ, (2.31) becomes a nonlinear Schrödinger D1 236 F. Winterberg · Planck Mass Plasma Vacuum Conjecture equation which can be derived from the Lagrange den- In the quantum equation of motion sity dF i 2 = [H,F], (2.41) ∗ h¯ ∗ dt h¯ L± = ih¯ϕ±ϕ˙± ∓ ( ϕ±)( ϕ±) 2m p (2.33) = −1 where H D1 H, the r.h.s. remains unchanged be- ∓ 2 1ϕ∗ ϕ − ϕ∗ ϕ ϕ∗ ϕ 2¯hcrp ± ± ∓ ∓ ± ± cause the Poisson bracket for any dynamical quantity 2 can be reduced to a sum of Poisson brackets for posi- with the Hamilton density tion and momentum, with the operator d/dt replaced by ∆ . The equation of motion is therefore changed into 2 1 = ± h¯ ( ϕ∗ )( ϕ ) H± ± ± i 2mp ∆ F = [H,F]. (2.42) (2.34) 1 h¯ 2 1 ∗ ∗ ∗ ∓ 2¯hcr ϕ±ϕ± − ϕ∓ϕ∓ ϕ±ϕ±. p 2 The Lagrange density now is With H± = H±dr one has the Heisenberg equation of 2 ∗ h¯ ∗ motion L± = ih¯ϕ±∆ ϕ± ∓ (D ϕ±)(D ϕ±) 1 2m 1 1 p (2.43) ih¯ψ˙± =[ψ±,H±], (2.35) ∓ 2 1ϕ∗ ϕ − ϕ∗ ϕ ϕ∗ ϕ . 2¯hcrp ± ± ∓ ∓ ± ± which agrees with (2.31) and shows that the classical 2 and quantum equations have the same form. For the ∗ † Variation with regard to ϕ according to particle number operator N± = ψ±ψ±dr one finds that ∂L ∂L ± − ± = ∂ϕ∗ D1 ∂( ϕ∗ ) 0 (2.44) ih¯N˙± =[N±,H±]=0, (2.36) ± D1 ± which shows that the particle numbers are conserved, leads to (2.40) replacing ϕ ∗, ϕ by ψ †, ψ. Variation permitting the permanent existence of negative masses. with regard to ϕ according to The foregoing can be carried over to the non- Archimedean formulation in each step. With the finite ∂L± ∂L± ∂L± − D1 − ∆1 = 0 (2.45) difference operators the commutation relations (2.32) ∂ϕ± ∂(D1ϕ±) ∂(∆1ϕ±) become leads to the conjugate complex equation. † ψ±(r),ψ±(|r |) = D(|r − r |), The momentum density conjugate to ϕ is (2.37) † † [ψ±(r),ψ±(r )] = ψ±(r)ψ (r ) = 0, ∂L± ∗ π± = = ih¯ϕ±, (2.46) ∂(∆1ϕ±) where D(|r − r|) is the generalized three-dimensional delta function and hence the Hamilton density (| − |)=δ(| − |), lim D r r r r (2.38) H± = π±∆1ϕ± − L± rp→0 and where ih¯ = ∓ (D1π±)(D1ϕ∓) 1 2mp D(|r − r|)=1. (2.39) D 1 − 2 1ϕ∗ ϕ − ϕ∗ ϕ π ϕ 2icrp ± ± ∓ ∓ ± ± (2.47) This changes (2.31) into the non-Archimedean form 2 2 2 = ± h¯ ( ϕ∗ )( ϕ ) ∆ ψ = ∓ h¯ 2ψ ± 2(ψ† ψ − ψ† ψ )ψ . D1 ± D1 ± ih¯ 1 ± D1 ± 2¯hcrp ± ± ∓ ∓ ± 2mp 2mp (2.40) 2 1 ∗ ∗ ∗ + 2¯hcr ϕ±ϕ± − ϕ∓ϕ∓ ϕ±ϕ±. p 2 F. Winterberg · Planck Mass Plasma Vacuum Conjecture 237 −1 ψ †ψ For the quantum equation of motion we have and applying to (2.51) D1 one obtains 2 . Insert- ing these results into (2.48) one obtains (2.40). This ih¯∆1ψ± =[ψ±,H±] (2.48) shows that the (non-Archimedean) classical and quantum equations have the same form. 2 = ψ , −1 h¯ 1ψ† ( ψ ) With the particle number operator ± D1 D1 ± D1 ± 2mp = −1ψ† ψ N± D ± ± (2.52) −1 2 1 † † † 1 + ψ±,±D 2¯hcr ψ± ψ± −ψ∓ ψ∓ ψ± ψ± . 1 p 2 one can show that With (2.37) one has for the first commutator ih¯∆ N± =[N±,H±]=0, (2.53) 1 ψ, −1 ψ† ( ψ) D1 D1 D1 establishing the permanent existence of negative = − ψ, −1 ψ† 2 ψ masses in the non- Archimedean formulation of the D1 D1 theory. − † = − 1 ψ ,ψ† 2 ψ (2.49) As before, ψ−ψ− can be treated as a c-number with D1 D1 † regard to the operators ψ+, ψ+ and vice versa. In these = − −1 2ψ (| − |) † † D D1 D r r terms ψ−ψ− (resp. ψ+ψ+) acts like an external potential. It is well known that for a nonrelativistic field the- = − 2ψ. D1 ory without interaction, but in the presence of an external potential the particle number is conserved. We In evaluating the second commutator we can take the therefore have only to show that this is also true if the ψ † ψ operator − − as a numerical (c-number) function nonlinear self-interaction term is included. In the com- † with regard to the operators ψ+, ψ+ and likewise the mutator it leads to integrals with integrands of the form † operator ψ+ψ+ as a numerical function with regard to † † † † † † † the operators ψ−, ψ−. For these terms we have expres- ψ ψψ ψ ψ ψ − ψ ψ ψ ψ ψ ψ sions in the integrand of the form = ψ†ψ† ψψψ† ψ + ψ†D(|r − r|)ψψ† ψ ψψ† ψ −ψ† ψψ = ψ(ψψ† −ψ† ψ)=ψD(|r−r|). − ψ† ψψ† ψψ†ψ (2.50) † † † † † −1 = ψ ψ ψ ψψ ψ + ψ D(|r − r |)ψ ψ ψ Applying to (2.50) the operator D 1 from the left, one obtains ψ. For the remaining terms one has − ψ† ψψ† ψψ†ψ ψψ† ψ ψ† ψ − ψ† ψ ψ† ψ ψ = ψ† ψψ†ψψ† ψ − ψ†D(|r − r|)ψψ† ψ = ψ† ψψ ψ† ψ + D(|r − r |)ψ ψ† ψ + ψ†D(|r − r|)ψψ† ψ − ψ†ψψ† ψψ†ψ − ψ† ψψ† ψψ = ψ† ψ[ψ†ψψ† ψ − ψ†ψψ†ψ] = ψ† ψψψ† ψ − ψ†ψψ† ψψ † † + ψ D(|r − r |)ψ ψ ψ + D(|r − r |)ψ ψ† ψ † † = ψ† ψ ψψ† ψ − ψ† ψψ − ψ D(|r − r |)ψψ ψ (2.54)

† with the last two terms in (2.54) canceling each other + D(|r − r |)ψ ψ ψ −1 by multiplication from the left with D1 . The first term = ψ† ψψD(|r − r|)+D(|r − r|)ψψ† ψ is zero as well, because it can be reduced to a term which would arise in the presence of an externally ap- = 2ψ† ψψD(|r − r|), (2.51) plied potential. 238 F. Winterberg · Planck Mass Plasma Vacuum Conjecture ∂ 3. Hartree-Fock Approximation 2 3 (v+ − v−)=−6c r (n+ − n−), (3.8) ∂t p For wave lengths large compared to the Planck ∂ n± + = , length it is sufficient to solve (2.31) nonperturbatively ∂ n± v± 0 with the Hartree-Fock approximation. For tempera- t 2 tures kT mpc , the Planck mass plasma is a two- where n± is a disturbance of the equilibrium density = / 3 component superfluid, with each mass component de- n± 1 2rp. Eliminating n± from (3.8) one obtains two scribed by a completely symmetric wave function. In wave equations: the Hartree-Fock approximation one has ∂2 2 2 † ∼ ∗ 2 (v+ + v−)=c (v+ + v−), ψ±ψ±ψ± = 2ϕ±ϕ±, ∂t2 (3.9) (3.1) 2 ψ† ψ ψ ∼ ϕ∗ ϕ ϕ , ∂ ∓ ∓ ± = ∓ ∓ ± (v+ − v−)=3c2 2(v+ − v−), ∂t2 whereby (2.31) becomes the two-component nonlinear Schrödinger equation: the first, for a wave propagating with c, with the oscillations of the positive and negative masses in phase,√ 2 ∂ϕ± h¯ while for the second the wave is propagating with 3c, = ∓ 2ϕ ± 2[ ϕ∗ ϕ − ϕ∗ ϕ ]ϕ . ◦ ih¯ ± 2¯hcrp 2 ± ± ∓ ∓ ± with the oscillations out of phase by 180 . The first ∂t 2mp wave is analogous to the ion acoustic plasma wave (3.2) while the second resembles electron plasma oscilla- With the Madelung transformation tions. From (3.4), (3.5) and (3.7) one obtains furthermore = ϕ∗ ϕ , n± ± ± two quantized, vortex solutions, for which (3.3) ih¯ ∗ ∗ − = ± +, n±v± = ∓ [ϕ± ϕ± − ϕ± ϕ±], v v 2mp n+ − n− = 0, (3.2) is brought into the hydrodynamic form ∂ rp v± 1 |v±| = vϕ = c , r > rp, (3.10) +(v±• )v± = − (U± + Q±), r ∂t mp (3.4) = 0 , r < r , ∂n± p + •(n±v±)=0, 1 1 rp 2 ∂ ± = − . t n 3 1 2rp 2 r where = 2 3( − ), The vortex core radius r = rp, where |v±| = c, is ob- U± 2mpc rp 2n± n∓ tained by equating U± with Q±. √ 2 2 (3.5) h¯ n± = − √ , Q± 4. Formation of a Vortex Lattice 2mp n±

ϕ± = A±eiS± , A± > 0, 0 ≤ S± ≤ 2π, In a frictionless fluid the Reynolds number is infinite. In the presence of some internal motion the flow (3.6) 2 h¯ is unstable, decaying into vortices. By comparison, n± = A±, v± = ± gradS±, mp the superfluid Planck mass plasma is unstable even in nh the absence of any internal motion, because with an v± • dr = ± , n = 0,1,2···. (3.7) equal number of positive and negative mass particles it m p can without the expenditure of energy by spontaneous For small amplitude disturbances of long wave symmetry breaking decay into a vortex sponge (resp. lengths one can neglect the quantum potential Q ± vortex lattice). against U±, and one obtains from (3.4) and (3.5) In nonquantized fluid dynamics the vortex core ra- ∂ dius is about equal a mean free path λ, where the ve- 2 3 (v+ + v−)=−2c r (n+)(n−), ∂t p locity reaches the velocity of sound, the latter about F. Winterberg · Planck Mass Plasma Vacuum Conjecture 239 equal to the thermal molecular velocity vt . With the 5. The Origin of Charge kinematic viscosity υ vt ,λ, the Reynolds number in the vortex core is Through their zero point fluctuations Planck mass particles bound in vortex filaments have a kinetic en- Re = vr/υ = vtλ/υ = 1. (4.1) ergy density by order of magnitude equal to

2 Interpreting Schrödinger’s equation as an equation ε ≈ mpc = hc¯ . 3 4 (5.1) with an imaginary quantum viscosity υQ = ih¯/2mp ∼ rp rp irpc, and defining a quantum Reynolds number By order of magnitude this is about equal the energy Q density g2, where g is the Newtonian gravitational field Re = ivr/υQ, (4.2) of a Planck mass particle mp at the distance r = rp. one finds with υ ∼ ir c that in the vortex core of a Because Q p √ quantum fluid (where v = c), ReQ ∼ 1. Because of this ∼ Gmp analogy one can apply the stability analysis for a lattice g = , (5.2) r2 of vortices in nonquantum fluid dynamics to a vortex p lattice in quantum fluid dynamics. For the two- dimen- one has sional Karman vortex street the stability was analyzed by Schlayer [16], who found that the radius r of the 0 Gm2 hc¯ vortex core must be related to the distance between 2 ≈ p = . g 4 4 (5.3) two vortices by rp rp

− The interpretation of this result is as follows: Through r 3.4 × 10 3. (4.3) 0 its zero-point fluctuations a Planck mass particle bound in a vortex filament becomes the source of virtual r = r = R R Setting 0 p and 2 , where is the radius of a phonons setting up a Newtonian type attractive force vortex lattice cell occupied by one line vortex, one has 2 = field with the coupling constant Gmp hc¯ . Charge is thus explained by the zero point fluctuations of the R 147. (4.4) Planck mass particles bound in vortices. The small- rp ness of the gravitational coupling constant is explained by the near cancellation of the kinetic energy coming No comparable stability analysis seems to have been from the positive and negative mass component of the performed for a three-dimensional vortex lattice made Planck mass plasma. Such a cancellation does not hap- up of vortex rings. The instability leading to the de- pen for fields not coupled to the energy momentum cay into vortices apparently arises from the disturbance tensor. Therefore, with the exception of the gravita- one vortex exerts on an adjacent vortex. At the distance tional coupling constant, all other coupling constants / R rp the velocity by a vortex ring is larger by the factor are within a few orders of magnitude equal tohc ¯ [18]. log(8R/rp), compared to the velocity of a line vortex at the same distance. With R/rp 147 for a line vortex 6. Maxwell’s and Einstein’s Equations lattice, a value of R/rp for a ring vortex lattice can be estimated by solving for R/rp the equation There are two kinds of transverse waves propagating through a vortex lattice, one simulating Maxwell’s 8R R/rp 147log , (4.5) electromagnetic and the other one Einstein’s gravita- rp tional wave. For electromagnetic waves this was shown by Thomson [19]. and one finds that [17] Let v = {vx,vy,vz} be the undisturbed velocity of the vortex lattice and u = {u ,u ,u } a small superim- / . x y z R rp 1360 (4.6) posed velocity disturbance. Only taking those disturbances for which divv =divu = 0, the x-component of 240 F. Winterberg · Planck Mass Plasma Vacuum Conjecture the equation of motion for the disturbance is Elimination of ϕ from (6.6) and (6.7) results in a wave equation for u ∂ux ∂(vx + ux) = −(vx + ux) ∂t ∂x −(1/c2)∂2u/∂t2 + 2u = 0. (6.8) ∂(vx + ux) − (vy + uy) (6.1) → ∂y In the continuum limit R rp, the microvelocity | | ϕ = ∂( + ) v has to be set equal to c. Then putting − ( + ) vx ux . −( / ) , = vz uz ∂ 1 2c H u E, (6.6) and (6.7) are Maxwell’s vac- z uum field equations. From the continuity equation divv = 0 one has Adding (6.1) and (6.2) and taking the average over x, y and z one has ∂vx ∂vy ∂vz vx + vx + vx = 0. (6.2) ∂x ∂y ∂z ∂ ∂ 2 ∂ ∂ ux = − vx − vxvy − vxvz , ( . ) ∂ ∂ ∂ ∂ 6 9a Subtracting (6.2) from (6.1) and taking the y-z average, t x y z one finds and similar ∂ ∂( ) ∂( ) ux = − vyvx − vzvx ( . ) ∂ ∂ ∂ 6 3a ∂u ∂v2 ∂v v ∂v v t y z y = − y − y z − y x , (6.9b) ∂t ∂y ∂z ∂x and similarly, by taking the x-z and x-y averages: ∂ ∂ 2 ∂ ∂ ∂u ∂(v v ) ∂(v v ) uz vz vzvx vzvy y = − x y − z y , (6.3b) = − − − . (6.9c) ∂t ∂x ∂z ∂t ∂z ∂x ∂y = ∂u ∂(v v ) ∂(v v ) With div u 0 this leads to z = − x z − y z . (6.3c) ∂t ∂x ∂y ∂2 ( )= . With divu = 0 one obtains from (6.3a – c) vivk 0 (6.10) ∂xi∂xk = vivk vkvi (6.4) For (6.9a – c) one can write Taking the x-component of the equation of motion, ∂ ∂ uk = − ( ). multiplying it by vy and then taking the y-z average, vivk (6.11) ∂t ∂xi and the y-component multiplied by vx and taking the x- z average, finally subtracting the first from the second Multiplying the vi component of the equation of mo- of these equations one finds tion with vk, and vice versa, its vk-component with vi, adding both and taking the average, one finds ∂ ∂u ∂u (v v )=−v2 y − x , (6.5) ∂t x y ∂x ∂y ∂ ∂ ∂ ( )=− 2 ui + uk . ∂ vivk v ∂ ∂ (6.12) 2 = 2 = 2 = 2 t xk xi where v vx vy vz is the average microvelocity square of the vortex field. 2 From (6.11) one has Putting ϕz = −vxvy/2v , (6.5) is just the z-component of ∂2u ∂ k = (v v ), (6.13) ∂ϕ ∂t2 ∂t∂x i k = 1 , i ∂ curlu (6.6) t 2 and from (6.12) ϕ = − / 2,ϕ = − / 2 where x vyvz 2v y vzvx 2v . The equations ∂2 ∂ ∂ ∂2 ∂2 (6.3a – c) then take the form 2 ui uk 2 uk (viv )=−v + = −v , ∂x ∂t k ∂x ∂x ∂x2 ∂x2 ∂u i k i i i = −2v2curlϕ. (6.7) ∂t (6.14) F. Winterberg · Planck Mass Plasma Vacuum Conjecture 241

Fig. 1. Deforma- tion of the vortex lattice for electromagnetic and gravitational waves.

the latter because div u = 0. Eliminating vivk from In (6.19) and (6.20) ε =(εx,εy,εx) is the displacement (6.13) and (6.14) and putting as before v 2 = c2, ﬁnally vector, which is related to the velocity disturbance vec- results in tor u by

2 2 ∂ε ∂ uk ∂ uk = . = c2 . (6.15) u (6.21) ∂ 2 ∂ 2 ∂t t xi In an elastic medium, transverse waves obey the wave or equation 2 1 ∂ u ∂2ε 2u − = 0. (6.16) 2 1 2 ∂ 2 ε − = 0. (6.22) c t c2 ∂t2 The line element of a linearized gravitational wave Because of (6.21), this is the same as (6.16). From the propagating into the x1-direction is condition div u = 0 and (6.21) then also follows div ε =0. 2 = 2 + 2 + + 2, ds ds0 h22dx2 2h23dx2dx3 h33dx3 (6.17) For a transverse wave propagating into the x-direction, εx = ε1 = 0. The condition div ε then leads to ( ≡ ) where x x1 ∂ε ∂ε 2 + 3 = ε + ε = 0. (6.23) ∂x ∂x 22 33 h22 = −h33 = f (t − x/c), h23 = g(t − x/c) (6.18) 2 3 The same is true for a gravitational wave putting with f and g two arbitrary functions, and ds 2 the line 0 2εik = hik. element in the absence of a gravitational wave. A de- Figure 1 illustrates the deformation of the vortex lat- formation of an elastic body can likewise be described tice for electromagnetic and gravitational waves. by a line element The totality of the vortex rings can be viewed as a fluid obeying an exactly nonrelativistic equation of 2 = 2 + ε , ds ds0 2 ikdxidsk (6.19) motion. It therefore satisfies a nonrelativistic continuity equation which has the same form as the equation where for charge conservation 1 ∂ε ∂ε ∂ρ ε = i + k . (6.20) e + = , ik ∂ ∂ divje 0 (6.24) 2 xk xi ∂t 242 F. Winterberg · Planck Mass Plasma Vacuum Conjecture ρ where e and je are the electric charge and current den- It has been argued that any kind of an aether the- sity. Because the charges are the source of the electro- ory with a compressible aether should lead to lon- magnetic field, one has gitudinal compression waves. Since the Planck mass plasma as a compressible medium falls into this cate- = πρ , divE 4 e (6.25) gory, we must explain why longitudinal waves are not observed. The transverse vortex lattice waves describ- and in order to satisfy (6.24), a term must be added ing Maxwell’s electromagnetic and Einstein’s gravita- to the Maxwell equation if charges are present, which tional waves, though, require longitudinal compression thereby becomes waves to couple the vortices of the vortex lattice. For 1 ∂E 4π this coupling to work there must be longitudinal waves + j = curlH. (6.26) at least in the wave length range r < λ < R. For short c ∂t c e p wave lengths approaching the Planck length, the wave As the hydrodynamic form (6.6) shows, Maxwell’s equations are modified by the quantum potential. In equation (−1/c)∂H/∂t = curlE, is purely kinematic the limit in which the quantum potential dominates, the and unchanged by the presence of charges. Finally, be- equation of motion (3.4) is cause of div ϕ = 0, it follows that div H = 0. A gravitational wave propagating into the x-direc- ∂v± 1 = − Q±. (6.34) tion obeys the equation ∂t mp 2 2 1 ∂ ∂ With the help of the expression for Q± and the lin- − + h = 0, i,k = 1,2,3,4. (6.27) c2 ∂t2 ∂x2 ik earized continuity equation one finds that ∂2 2 By a space-time coordinate transformation it can be v± h¯ 4 = − v±, (6.35) brought into the form ∂ 2 2 t 4mp ψk = , , = , , , i 0 i k 1 2 3 4 (6.28) leading to the nonrelativistic dispersion relation for the free Planck masses with the subsidiary (gauge) condition ω = ± 2/ . ∂ψk hk¯ 2mp (6.36) i = 0, (6.29) ∂ k x And with the inclusion of the quantum potential the ψ k = k − ( / )δ k first equation of (3.9) is modified as follows: where i hi 1 2 i h. In the presence of matter, the gravitational field ∂2 2 equation must have the form 2 2 h¯ 4 (v+ + v−)=c (v+ + v−) − (v+ + v−), ∂ 2 2 t 4mp ψk = κΘk κ = const, (6.30) i i (6.37) where Θ k is a four-dimensional tensor. Because of i possessing the dispersion relation (6.29), it obeys the equation

1/2 ∂Θ k h¯2 i = 0. (6.31) ω = c2k2 + k4 (6.38) ∂xk 4mp Θ k As it was shown by Gupta [20], splitting i in its mat- first derived by Bogoliubov [23] for a dilute Bose gas. k k ter part Ti and gravitational field part ti , The influence of the quantum potential on the propa- gation of the waves for wave lengths larger than the Θ k = k + k, i Ti ti (6.32) Planck length can normally be neglected, but this is not the case if these waves propagate through a vortex (6.30) can be brought into Einstein’s form lattice. There the influence of the quantum potential is 1 estimated as follows: The quantum potential is largest Rik − gikR = κTik. (6.33) near the vortex core, where according to (3.10) for 2 F. Winterberg · Planck Mass Plasma Vacuum Conjecture 243 √ > / r rp 2 the particle number√ density becomes zero. 4 ≈ ( / )4 = / 4 There we may put 2 rp 4 rpp, whereby

2 2 h¯ 4 c ± ±. v 2 v (6.39) 4mp rp If the vortex lattice consists of ring vortices with a ring radius R, and if it is evenly spaced by the same distance, the average number density of Planck masses bound in the vortices under these conditions is of the 3 order of (R/rp)/R . The average effect of the quantum potential on the vortex lattice is then obtained by mul- 3/ 2 tiplying (6.39) with the factor rp R rp, whereby one has for the average over the vortex lattice Fig. 2. A pole-dipole particle executes a circular motion h¯2 c 2 around its center of mass S. 4 = ω2 , ω = / . v± v± 0 v± 0 c R (6.40) 4mp R bound in the vortex core are the source of a gravitational charge with Newton’s coupling constant, but With (6.40) the wave equation (6.37) is modified as whereas the gravitational interaction energy of two follows: positive masses is negative, it is positive for the inter- ∂2 action of a positive with a negative mass. Adding the 2 2 2 (v+ +v−)=c (v+ +v−)−ω (v+ +v−), (6.41) ∂ 2 0 mass m of the small positive gravitational interaction t + + = + + − = − energy to mv , whereby m mv m and m mv , having the dispersion relation one obtains a configuration which has been called a pole-dipole particle. The importance of this is that it ω c + − = . (6.42) can simulate Dirac spinors [22]. With m > |m |, one 2 + − + k 1 − (ω0/ω) has m −|m |m . The center of mass S of this two body configuration is still on the line connecting m + ω = ω = / − + − It has a cut-off at 0 c R, explaining why there with m , but not located in between m and m , with λ > are no longitudinal waves for wave lengths R. the distance from S to the position halfway in between Accordingly, these longitudinal waves could be de- + − + m and m equal to rc (see Figure 2). If m is sepa- tected only above energies corresponding to the length rated by the distance r from m−, and if the distance of R ∼ 10−30 cm, that is above the GUT energy of + m from the center of mass is rc − r/2 ≈ rc, then be- ∼ 1016 GeV. + ∼ − cause of m m = |m | one has r rc. Conservation of the center of mass requires that 7. Dirac Spinors + − m rc = |m |(rc + r). (7.3) A vortex ring of radius R has under elliptic deforma- The angular momentum of the pole-dipole particle is tions the resonance frequency [21] = + 2 −| −|( + )2 ω, Jz m rc m rc r (7.4) crp ω = , (7.1) ω v R2 where is the angular velocity around the center of mass. With m = m+ −|m−| and p = m+r =∼ |m−| =∼ and for the energy of its positive and negative mass mrc, where m is the mass pole and p the mass dipole component with p directed from m+ to m−, one finds with the help of (7.3) that 2 2 rp 2 h¯ωv = ±mpc = ±mvc . (7.2) = − + ω = − . R Jz m rrc pv (7.5)

12 In the limit v = c, one has For R/rp 1360 one hash ¯ω 5 × 10 GeV. The zero point fluctuations of the Planck mass particles Jz = −mcrc. (7.6) 244 F. Winterberg · Planck Mass Plasma Vacuum Conjecture − The angular momentum is negative because m is sep- For the electron mass one would have to set R/rp ∼ arated by a larger distance from the center of mass 5000 instead of R/rp ∼ 1360. This shows that the value + than m . for R/rp obtained from a rough estimate of the vortex Applying the solution of the well-known nonrela- lattice dimensions is still very tentative. tivistic quantum mechanical two- body problem with With the gravitational interaction of two counter-ro- Coulomb interaction to the pole-dipole particle with tating masses reduced by the factor γ 2 = 1 − v2/c2, Newtonian interaction, one can obtain an expression where v is the rotational velocity [24], the mass for for m. For the Coulomb interaction, the groundstate en- the vortices in counter-rotating motion is much smaller. ergy is By order of magnitude one has γ mvRc ∼ h¯, and hence γ ∼ R/r , whereby instead of (7.10) one has ∗ 4 p = −1 m e , W0 2 (7.7) 8 2 h¯ m/mp ≈ (rp/R) . (7.13) ∗ where m is the reduced mass of the two-body sys- / =∼ × 3 ∼ × 2 For R rp 5 10 , this leads to a mass of 2 tem, with the potential energy −e /r for two charges −2 ± 10 eV, making it a possible candidate for the neu- e, of opposite sign. By comparison, the gravitational trino mass. potential energy of two masses of opposite sign is + +| −|/ ∼ + | ±|2/ The result expressed by (7.10) amounts to Gm m r = G mv r instead, and one thus has 2 →− | ±|2 a computation of the renormalization constant, to make the substitution e G mv . κ = which in this√ theory is finite and equal to The reduced mass is 6 mp 1 − 1/ 2 rp/R , where m = mp − κ is the 1 1 1 1 1 m = + = + =∼ , (7.8) difference between two very large masses. ∗ + − + | −| | ±|2 m m m m m mv In the pole-dipole particle configuration the spin angular momentum is the orbital angular momentum of and by putting W = mc2, one finds from (7.7) that 0 the motion around the center of mass located on the √ + − ( / )| ±|3 line connecting m with m . The rules of quantum me- 1 2 mv m = . (7.9) chanics permit radial s-wave oscillations of m+ against m2 p m−. They lead to the correct angular momentum quan- Because of (7.2) this is tization for the nonrelativistic pole-dipole particle con- √ figuration, as can be seen as follows: From Bohr’s an- 6 ∗ m/mp =(1/ 2)(rp/R) . (7.10) gular momentum quantization rule m rvv = h¯, where ∗ v = rvω, one obtains by inserting the values for m and 2 The Bohr radius for the hydrogen atom is rv that mc = −(1/2)h¯ω, and because of ω = c/rc that = −( / ) | | = / 2 mrcc 1 2 h¯, or that rc h¯ 2mc. Inserting this = h¯ , result into (7.6) leads to J =(1/2)h¯. Therefore, even in rB ∗ (7.11) z m e2 the nonrelativistic limit the correct angular momentum which by the substitution for e2 and m∗ becomes quantization rule is obtained, the only one consistent with Dirac’s relativistic wave equation. For the mu- √ √ 2 = / | ±| = / / 2 . tual oscillating velocity one finds v/c = |m±|/m = rv h¯ 2 mv c rp 2 R rp (7.12) v p 4 rp/R 1 , showing that a nonrelativistic approxi- With the above computed value R/rp = 1360 one finds mation appears well justified. that m ∼ 1 GeV, about equal the proton mass, and r v ∼ To reproduce the Dirac equation, the velocity of the − 2 × 10 27 cm. double vortex ring, moving as an excitonic quasiparti- This result can in a very qualitative way also be cle on a circle with radius rc, must become equal to the obtained as follows: Equating the positive gravita- velocity of light. Because rc = h¯/2mc one has in the tional interaction energy with the rest mass one has limit m → 0, rc → ∞ and a straight line for the trajec- mc2 = G|m |2/r, and from the uncertainty principle tory. ∼ v |mv|rc = h¯. Eliminating r from these two relations As shown above the presence of negative masses 3 one obtains m/mp =(|mv|/mp) , and with |mv| = leads to a “Zitterbewegung,” by which a positive mass 2 6 mp(rp/R) , m/mp =(rp/R) . is accelerated. According to Bopp [25], the presence of F. Winterberg · Planck Mass Plasma Vacuum Conjecture 245 negative masses can be accounted for in a generalized That this is the equation of motion of a pole-dipole dynamics where the Lagrange function also depends particle can be seen by writing it as follows: on the acceleration. The equations of motion are there dPα derived from the variational principle = 0, ds (7.24) δ L(qk,q˙k,q¨k)dt = 0, (7.14) 3 Pα = k − k u˙2 uα + k u¨α . 0 2 1 v 1 or from By the angular momentum conservation law δ Λ(xα ,uα ,u˙α )ds = 0, (7.15) dJαβ = 0, (7.25) 1/2 ds where uα = dxα /ds,˙uα = duα /ds,ds = 1 − β 2 dt, β = v/c, xα =(x1,x2,x3,ict), and where L = where / 2 1 2 2 Λ 1 − β . With the subsidiary condition F = uα = =[ , ] +[ , ] , −c2, the Euler-Lagrange equation for (7.15) is Jαβ x P αβ u p αβ (7.26) d ∂Λ + λF d ∂Λ ∂Λ the mass dipole moment is equal to − − = 0 (7.16) ds ∂uα ds ∂u˙α ∂xα pα = −k1u˙α . (7.27) with λ a Lagrange multiplier. In the absence of exter- 2 = , = , , nal forces, Λ can only depend onu ˙ α . The most simple For a particle at rest with Pk 0 k 1 2 3, one obtains assumption is a linear dependence Jkl =[u,p]kl = uk pl − u1 pk, (7.28) 2 Λ = −k0 − (1/2)k1u˙α , (7.17) which is the spin angular momentum. whereby (7.16) becomes The energy of a pole-dipole particle at rest, for = γ d which u4 ic , is determined by the fourth component (2λuα + k u¨α )=0 (7.18) of (7.24): ds 1 or = = − 3 2 γ. P4 imc i k0 k1u˙v c (7.29) ˙ 2 2λuα + 2λu˙α + k1u¨α = 0. (7.19) From (7.24) and (7.27) for P = 0, k = 1, 2, 3, one Differentiating the subsidiary condition u2 = −c2, one k α obtains for the dipole moment p: has 2 ··· 3 uαu˙α = 0, uαu¨α + u˙α = 0, uαuα + 3˙uαu¨α = 0, (7.20) p = k − k u˙2 r , (7.30) 0 2 1 v c by which (7.19) becomes and because of (7.29) one has 3 d 2 −2λ˙ − 3k u˙u¨α = −2λ˙ − k (u˙α )=0. (7.21) 1 2 1 ds = mrc . p γ (7.31) This equation has the integral (summation over v) With u = γv, one obtains for the spin angular momen- λ = − ( / ) 2, 2 k0 3 2 k1u˙v (7.22) tum where k appears as a constant of integration. By in- 0 J = −pu = mvr =∼ −mcr , (7.32) serting (7.22) into (7.18) the Lagrange multiplier is z c c eliminated and one has which is the same as (7.6). d 3 2 For the transition to wave mechanics, one needs k − k u˙ uα + k u¨α = 0. (7.23) ds 0 2 1 v 1 the equation of motion in canonical form. From L = 246 F. Winterberg · Planck Mass Plasma Vacuum Conjecture Λds/dt one obtains, by separating space and time parts with the mass given by (using units in which c = 1): 2 2 2 m = k0 − (1/2k1)(1 − k1)(1 − v )[s − (s · v) ]. (7.42) 1 2 2 1/2 L = − k + k u˙α (1 − v ) , 0 2 1 Higher particle families result from internal excited states of the pole-dipole configuration. They can also · 2 (7.33) 2 1 2 v v be obtained by Bopp’s generalized mechanics putting u˙α = v˙ + , [(1 − v2)1/2]4 (1 − v2)1/2 2 Λ = − f (Q), Q = u˙α , (7.43) where L = L(r,˙r,¨r). From where f (Q) is an arbitrary function of Q which de- ∂L d ∂L ∂L pends on the internal structure of the mass dipole. P = − , s = (7.34) ∂v dt ∂v˙ ∂v˙ With (7.43) one has for (7.16) one can compute the Hamilton function d d [ f (Q) − 4Qf (Q)]uα + 2 [ f (Q)]u˙α = 0, ds ds H = v · P + v˙ · s − L. (7.35) (7.44) From s = ∂L/∂v˙ one obtains where k (v · v˙)v s = −√ 1 v˙ + , = − ( )duα 3 − 2 pα 2 f Q (7.45) 1 − v2 1 v ds (7.36) √ 3 1 − v2 is the dipole moment. For the simple pole-dipole parti- v˙ = − [s − (v · s)v], cle one has according to (7.17) f (Q)=k0 +(1/2)k1Q, k1 whereby pα = −k1u˙α as in (7.27). Instead of (7.36) by which, with the help of (7.33),v ˙ ·s can be expressed one now has in terms of v and s. In these variables the angular mo- f (Q) (v · v˙)v mentum conservation law (7.25) assumes the form s = −2√ v˙ + , 3 − 2 1 − v2 1 v × + × = , r P v s const (7.37) √ 3 (7.46) 1 1 − v2 v˙ = − [s − (v · s)v]. with the vector s equal the dipole moment. For the 2 f (Q) Hamilton function (7.35) one then finds Computation ofv ˙ ·s from both of these equations leads 2 1/2 H = v · P + k0(1 − v ) to the identity (7.38) 2 3/2 2 2 ( )2 = =( − 2)[ 2 − ( · )2], − (1/2k1)(1 − v ) [s − (s · v) ]. 4Qf Q R 1 v s v s (7.47) Putting from which the function Q = Q(R) can be obtained and by whichv ˙ can be eliminated from H: ∂ h¯ 2 1/2 P = , v = α, (1 − v ) = α4, (7.39) i ∂r H = v · P+ v˙ · s − L = v · P + 1 − v2F(R), (7.48) α = {α,α } where 4 are the Dirac matrices, one finally where obtains the Dirac equation F(R)= f (Q) − 2Qf(Q). (7.49) h¯ ∂ψ + Hψ = 0, (7.40) i ∂t For the wave mechanical treatment of this problem it is convenient to use the four-dimensional representation where by making a canonical transformation from the vari- ( , , ) ( , ) H = α1P1 + α2P2 + α3P3 + α4m, ables v v0;s s0 to uα pα : (7.41) · + + = Φ( , , ) αµ αv + αvαµ = 2δµv s dv s0dv0 uαdpα d v vα pα (7.50) F. Winterberg · Planck Mass Plasma Vacuum Conjecture 247 ψ with the generating function ψ = 0 , tanhα = v, (7.61) sinhα v0 Φ = √ (v · p+ ip4) (7.51) the wave equation becomes − 2 1 v ∂2 M2 and where v0, s0 are superfluous coordinates. Ex- − − 1 − ψ = G(ε coshα)ψ , (7.62) ∂α2 2 0 0 pressed in the new variables, one has sinh α where 1 2 R = − Mαβ, Mαβ = uα pβ − uβ pα . (7.52) 2 M2 =(v × s)2 With Pα = {P,iH}, (7.48) is replaced by 2 (7.63) 1 ∂ ∂ 1 ∂ = − sinθ − , 2 θ ∂θ ∂θ 2 θ ∂φ 2 K = uα Pα + −uα F(R)=0. (7.53) sin sin having the eigenvalues j( j + 1), where j is an integer. Because u2 = −1 and uα pα = 0, the superfluous coor- α The wave equation, therefore, finally becomes dinates vo and so can be eliminated. Putting d2ψ h¯ ∂ h¯ ∂ 0 = V(α)ψ Pα = , pα = , (7.54) dα2 0 i ∂xα i ∂uα (7.64) ( + ) = + j j 1 − (ε α) ψ . one obtains the wave equation 1 G cosh 0 sinh2 α h¯ ∂ Kψ ≡ uα , + F(R) ψ(x,u)=0, (7.55) The eigenvalues can be obtained by the WKB method, i ∂xα with the factor j( j + 1) replaced by ( j + 1/2)2 to account for the singularity at α = 0. The eigenvalues are where then determined by the equation ∂ ∂ 1 2 h¯ R = − Mαβ, Mαβ uα − uβ . (7.56) 1 a2 1 2 i ∂uβ ∂uα = − (α) α = + , = , , ,··· j π V d n n 0 1 2 (7.65) a1 2 For P = 0, the wave function has the form with −iεt/h¯ ψ(x,u)=ψ(u)e (7.57) 2 ( j + 1/2) V(α)=1 + − G(ε coshα). (7.66) with the wave equation for ψ(u) sinh2 α ε Of special interest are the cases where j = −1/2, be- F(R)ψ(u)= √ ψ(u) (7.58) 1 − v2 cause they correspond to the correct angular momentum quantization rule for the Zitterbewegung. For j = or, if G is the inverse function of F: −1/2 one simply has ε Rψ(u)=G √ ψ(u). (7.59) V(α)=1 − G(ε coshα). (7.67) 1 − v2 To obtain an eigenvalue requires a finite value of = From the condition uα pα 0 follows that the phase integral (7.65). The function G(x), (x = ε α) ∂2 cosh , must therefore qualitatively have the form 2 R = −h¯ . (7.60) of a parabola cutting the line G = 1 at two points x1, ∂uα2 x2, in between which G(x) > 1. One can then distin- ε With (θ, φ spherical polar coordinates) guish two limiting cases. First if 1 and a second if ε 1. In both cases one may approximate (7.65) as uα =[sinhα sinθ cosφ, follows: ∼ sinhα sinθ sinφ,sinhα cosθ,coshα], J = (1/π) −V(α)(α2 − α1). (7.68) 248 F. Winterberg · Planck Mass Plasma Vacuum Conjecture In the first case α 1 and one has   x x2 2x ε2 α =∼ ln + − 1 ≈ ln − + ···, ε ε2 ε 4x2 (7.69) hence x x2 − x2 ε2 α − α =∼ 2 + 2 1 + ···. 2 1 ln 2 2 (7.70) x1 x1x2 4

In the second case one has = (ε) Fig. 3. Qualitative form of the phase integral J J to √ x 1/2 determine the number of families and their masses. α =∼ − , 2 ε 1 (7.71) One may ask what kind of mass distributions gen- or if x/ε 1 simply erate half integer spin quantization. The answer to this question is that noninteger angular momentum quan- 2x tization is possible if the rotational motion is accom- α =∼ , (7.72) ε panied by a simultaneous time-dependent deformation of the positive-negative mass distribution. A superpo- hence sition of a rotational motion with a simultaneous pe- 2 / / riodic deformation can for example, occur in rotating α − α =∼ (x1 2 − x1 2). 2 1 ε 2 1 (7.73) molecules if the rotation is accompanied by a time- dependent deformation. As it was demonstrated by ε One, therefore, sees that the phase integral has for Delacretaz et al. [26], it there can lead to half-integer = + ε 2 ε = 1 the√ form J a b , but for 1 the form J angular momentum quantization. The same must be / ε (ε) a . The J curve can for this reason cut twice the possible for the positive-negative mass vortex config- = / ( = ) = / ( = ) lines J 1 2 n 0 , and J 3 2 n 1 . uration. In Fig. 3, we have adjusted the phase integral to ac- ε = 2 count for the electron, muon and tau, where mc , 8. Finestructure Constant with m the electron mass. The fact that the mass ratio of the tau and muon are so much smaller than the According to Wilczek [27] the ratio of the baryon mass ratio of the muon and electron suggests that both mass m to the Planck mass mp can be expressed in the muon and tau result from cuts of the line J = 3/2. terms of the finestructure constant α at the unification Because of the proximity on the J = 3/2 line, it is un- energy of the electroweak and strong interaction: likely that the phase integral would cut the line J = 5/2 √ m − k or higher. Since for large values of ε,J ∝ 1/ ε, it fol- = e a , (8.1) m lows that there must be one more eigenvalue for which p J = 1/2, which from the position of the first three fami- where k = 11/2π is a calculable factor computed from ∼ lies is guessed to be around 80,000 mc2 = 40 GeV.This the antiscreening of the strong force. One thus has result suggests that there are no more than four parti- π 1 = 2 mp . cle families. In the framework of the proposed model a α log (8.2) more definite conclusion has to await a determination 11 m of the structure function f (Q) for the mass distribu- With the help of (7.10) one can then compute α [17]: tion of the exciton made up from the positive-negative 1 12π R mass vortex resonance. The finiteness of the number = log . (8.3) α 11 r of possible families is here a consequence of Bopp’s p nonlinear dynamics involving negative masses, not as For R/rp = 1360, one finds that 1/α = 24.8, in sur- in superstring theories, where it results from a topolog- prisingly good agreement with the empirical value, ical constraint. 1/α = 25. F. Winterberg · Planck Mass Plasma Vacuum Conjecture 249 9. Quantum Mechanics and Lorentz Invariance where ρ(r,t) are the sources of this field. For a body in static equilibrium, at rest in the distinguished reference According to Einstein and Hopf the friction force system for which the sources are those of the body it- acting on a charged particle moving with the velocity self, one has v through an electromagnetic radiation field with a frequency dependent spectrum f (ω) is given by [28] 2Φ = −4πρ(r). (9.4) ω d f (ω) if set into absolute motion with the velocity v along F = −const f (ω) − v. (9.1) 3 dω the x-axis, the coordinates of the reference system at rest with the moving body are obtained by the Galilei This force vanishes if transformation:

f (ω)=constω3. (9.2) x = x − vt, y = y, z = z, t = t, (9.5)

It is plausible that this is universally true, not only transforming (9.3) into for electromagnetic interactions. But now, the spec- 2 2 2 trum (9.2) is not only frictionless, but it is also Lorentz 1 ∂ Φ 2v ∂ Φ v2 ∂ Φ − + + 1 − invariant. c2 ∂t2 c2 ∂x∂t c2 ∂x2 (9.6) In the Planck mass plasma the zero point energy re- 2 2 ∂ Φ ∂ Φ sults from the “Zitterbewegung” caused by the inter- + + = 4πρ r t . ∂y2 ∂z2 action of positive with negative Planck mass particles. To relax it into a frictionless state as is the case for a After the body has settled into a new equilibrium in superfluid, the spectrum must assume the form (9.2). which ∂/∂t = 0, one has instead of (9.4) With 4πω2dω modes of oscillation in between ω and ω +dω the energy for each mode must be proportional v2 ∂2Φ ∂2Φ ∂2Φ to ω to obtain, as in quantum mechanics, the ω 3 de- 1 − + + = −4πρ(x,y,z). 2 ∂ 2 ∂ 2 ∂ 2 pendence (9.2). Because the spectrum (9.2) is gener- c x y z ated by collective oscillations of the discrete Planck (9.7) mass particles, it has to be cut off at the Planck fre- ω = / = / Comparison of (9.7) with (9.4) shows that the l.h.s. quency p c rp 1 tp, where the zero point energy 2 of (9.7) is the same it one sets Φ = Φ and dx = is equal to (1/2)h¯ωp =(1/2)mpc . It thus follows that − 2/ 2 the zero point energy of each mode with a frequency dx 1 v c . This implies a uniform contraction of − 2/ 2 ω < ωp must be E =(1/2)h¯ω. A cut-off of the zero the body by the factor 1 v c , because the sources point energy at the Planck frequency destroys Lorentz within the body are contracted by the same factor, invariance, but only for frequencies near the Planck whereby the r.h.s. of (9.7) becomes equal to the r.h.s. frequency and hence only at extremely high energies. of (9.4). Since all clocks can be viewed as light clocks The nonrelativistic Schrödinger equation, in which the (made up from rods with light signals reflected back zero point energy is expressed through the kinetic en- and forth along the rod), the length contraction leads to ergy term −(h¯ 2/2m) 2ψ, therefore remains valid for slower going clocks, going slower by the same factor. masses m < mp, to be replaced by Newtonian mechan- Thus using contracted rods and slower going clocks as ics for masses m > mp. measuring devices, (9.3) is Lorentz invariant. A cut-off at the Planck frequency generates a dis- With the repulsive quantum force having its cause tinguished reference system in which the zero point in the zero point energy and with the zero point energy energy spectrum is isotropic and at rest. In this dis- Lorentz invariant, it follows that the attractive electric tinguished reference system the scalar potential from force is balanced by the repulsive quantum force al- which the forces are to be derived satisfies the inho- ways in such a way that Lorentz invariance appears to mogeneous wave equation be true. Departures from Lorentz invariance could then only be noticed for particle energies near the Planck 1 ∂2Φ energy, where Lorentz invariance is violated through − + 2Φ = −4πρ(r,t), (9.3) c2 ∂t2 the cut-off of the zero point energy spectrum. 250 F. Winterberg · Planck Mass Plasma Vacuum Conjecture 10. Gauge Invariance According to (10.2) and (10.6) Φ and A shift the phase of a Schrödinger wave by In Maxwell’s equations the electric and magnetic fields can be expressed through a scalar potential Φ e t2 e δϕ = Φdt, δϕ = A · ds. (10.9) and a vector potential A: h¯ t1 hc¯ 1 ∂A The corresponding expressions for a gravitational field E = − − gradΦ, H = curlA. (10.1) c ∂t can be directly obtained from the equivalence princi- ∂ /∂ ω E and H remain unchanged under the gauge transfor- ple [29]. If v t is the acceleration and the angular mation of the potentials velocity of the universe relative to a reference system assumed to be at rest, the inertial forces in this system 1 d f are Φ = Φ − , A = A + gradf , (10.2) c dt ∂v Φ F = m + ω˙ × r − ω × (ω × r) − r˙ × 2ω . (10.10) where f is called the gauge function. Imposing on ∂t and A the Lorentz gauge condition For (10.10) we write 1 ∂Φ + divA = 0, (10.3) c ∂t 1 F = m Eˆ + v × Hˆ , (10.11) the gauge function must satisfy the wave equation c

1 ∂2 f where − + 2 f = 0. (10.4) c2 ∂t2 ∂v Eˆ = + ω˙ × r − ω × (ω × r), In an electromagnetic ﬁeld the force on a charge e is ∂t (10.12) Hˆ = −2cω . 1 F = e E + v × H With c 1 ∂A 1 ˙ = e − − gradΦ + v × curlA . curl(ω × r)=2ω˙ c ∂t c (10.13) (10.5) div(−ω × (ω × r)) = 2ω 2,

By making a gauge transformation of the Hamilton one has operator in the Schrödinger wave equation, the wave 1 ∂Hˆ function transforms as divHˆ = 0, + curlEˆ = 0. (10.14) c ∂t ie ψ = ψ exp f , (10.6) hc¯ Eˆ and Hˆ can be derived from a scalar and vector potential leaving invariant the probability density ψ ∗ψ. To give gauge invariance a hydrodynamic interpre- 1 ∂Aˆ Eˆ = − − gradΦˆ , Hˆ = curlAˆ . (10.15) tation, we compare (10.5) with the force acting on a c ∂t test body of mass m placed into the moving Planck mass plasma. This force follows from Euler’s equation Applied to a rotating reference system one has and is 1 Φˆ = − (ω × r)2, Aˆ = −c(ω × r), (10.16) dv ∂v v2 2 F = m = m + grad − v × curlv . (10.7) dt ∂t 2 or Complete analogy between (10.5) and (10.7) is estab- v2 Φˆ = − , Aˆ = −cv. (10.17) lished if one sets 2 m 2 mc Φ = − v , A = − v. (10.8) m/e 2e e Apart from the factor this is the same as (10.8). F. Winterberg · Planck Mass Plasma Vacuum Conjecture 251 √ For weak gravitational fields produced by slowly mass M is GM and the gravitational field generated moving matter, Einstein’s linearized gravitational field by M equations permit the gauge condition (replacing the √ Lorentz gauge) GM g = . (11.1) r2 4 ∂Φˆ + divAˆ = 0, The force exerted by g on another mass m (of charge c ∂t √ (10.18) Gm), which like M is composed of masses mp bound ∂Aˆ = 0 in vortex filaments, is ∂t √ GmM F = Gm × g = , (11.2) with the gauge transformation for Φˆ and Aˆ r2

Φˆ = Φˆ , but because the vortex lattice propagates tensorial waves, simulating those derived from Einstein’s grav- (10.19) itational field equations, the gravitational field appears Aˆ = Aˆ + grad f , to be tensorial in the low energy limit. where f has to satisfy the potential equation The equivalence of the gravitational and inertial masses can most easily be demonstrated for the limit- 2 f = 0. (10.20) ing case of an incompressible Planck mass plasma. To prove the principle of equivalence in this limit, we use For a stationary gravitational field the vector potential the equations for an incompressible frictionless fluid changes the phase of the Schrödinger wave function dv according to divv = 0, ρ = −grad p. (11.3) dt im ψ = ψexp f , (10.21) Applied to the positive mass component of (11.3), ρ = = / 3 hc¯ we have nmp, n 1 2rp. In the limit of an incompressible fluid, the pressure p plays the role of a leading to a phase shift on a closed path Lagrange multiplier, with which the incompressibility condition divv = 0 in the Lagrange density func- m m δϕ = − Aˆ · ds = v · ds. (10.22) tion has to be multiplied [37]. The force resulting hc¯ h¯ from a pressure gradient is for this reason a constraint force. The same is true for the inertial forces in general relativity, where they are constraint forces imposed 11. Principle of Equivalence by curvilinear coordinates in a noninertial reference system. According to (5.1), Newton’s law of gravitational The inertial force ρdv/dt in Euler’s equation can be attraction, and with it the property of gravitational interpreted as a constraint force resulting from the in- mass, has its origin in the zero point fluctuations of teraction with the Planck masses filling all of space. the Planck mass particles bound in quantized vortex Apart from those regions occupied by the vortex fila- filaments. For the attraction to make itself felt, both ments, the Planck mass plasma is everywhere super- the attracting and attracted mass must be composed of fluid and must obey the equation Planck mass particles bound in vortex filaments. The gravitational field generated by a mass M, different curlv = 0. (11.4) from mp, is the sum of all masses mp bound in vortex filaments, whereby fractions of mp are possible for With the incompressibility condition divv = 0, the so- an assembly consisting of an equal number of positive lutions of Euler’s equation for the superfluid regions and negative Planck masses, with the positive kinetic are solutions of Laplace’s equation for the velocity po- energy of the positive Planck masses different from tential ψ the absolute value of the negative kinetic energy of the negative Planck masses. The gravitational charge of the 2ψ = 0. (11.5) 252 F. Winterberg · Planck Mass Plasma Vacuum Conjecture where v = −gradψ. A solution of (11.5) is solely de- As Einstein’s equation of motion for a test body, termined by the boundary conditions on the surface of which is the equation for a geodesic in a curved four- the vortex filaments, as can be seen more directly in the dimensional space, equation (11.12) does not contain following way: With the help of the identity the mass of the test body. It is for this reason purely kinematic, giving a kinematic interpretation of inertial dv ∂v v2 = + − v× curlv (11.6) mass. dt ∂t 2 The integral on the r.h.s. of (11.12) must be extended v = one can obtain an equation for p by taking the diver- over all regions where curl 0. These are the re- gence on both sides of Euler’s equation gions occupied by vortex filaments. In the Planck mass plasma, the vortex core radius is the Planck length r p, p v2 with the vortex having the azimuthal 2 + = div(v × curlv) (11.7) ρ 2 vφ = c(rp/r)=0, r > rp. (11.13) Solving for p one has (up to a function of time depending on the initial conditions but which is otherwise of Using Stokes’ theorem for a surface cutting through no interest) the vortex core and having a radius equal to the core radius rp, one finds that at r = rp 2 p v div(v × curlv) = − − dr . (11.8) ρ 2 4π|r− r| 2c curlzv = . (11.14) rp Inserting this expression for p back into Euler’s equation, it becomes an integro-differential equation [38] A length element of the vortex tube equal to r p, makes 2 the contribution dv v div(v × curlv) = + dr . (11.9) dt 2 4π|r − r| div(v × curlv) 1 dr =∼ (v × curlv) · df, 4π|r − r| 4π|r − r| If the superfluid Planck mass plasma could be set into rp uniform rotational motion of angular velocity ω, one (11.15) would have ∼ π 2 where d f = 2 rp is the surface element for a length v = rω (11.10) rp of the vortex tube. Therefore, each such element r p contributes and hence 2 div(v × curlv) rpc 2 2 =∼ . | (v /2)| = rω , (11.11) π| − | dr | − | (11.16) rp 4 r r r r but because for a superfluid curlv = 0, the velocity The contribution to dv/dt of an element located at r field (11.10) is excluded. If set into uniform rotation, then is a superfluid rather sets up a lattice of parallel vortex filaments, possessing the same average vorticity as a dv r c2 = p (11.17) uniform rotation. If space would be permeated by such dt |r − r| an array of vortices, it would show up in an anisotropy 2 which is not observed. Other nonrotational motions Because of Gm = hc¯ and mprpc = h¯, this can be writ- 2 p leading to nonvanishing (v /2)-terms involve expan- ten as sions or dilations, excluded for an incompressible fluid. Therefore, the only remaining term on the r.h.s. of dv = Gmp , (11.18) (11.9) is the integral term. It is nonlocal and purely dt |r − r |2 kinematic. Through it a “field” is transmitted to the demonstrating the origin of the inertial mass in the position r over the distance |r − r |. One can therefore gravitational mass of all the Planck masses in the uni- write for the incompressible Planck mass plasma verse bound in vortex filaments. dv div(v × curlv) To obtain a value for the inertial mass of a body, its = dr. (11.12) dt 4π|r− r| interaction with all those Planck masses in the universe F. Winterberg · Planck Mass Plasma Vacuum Conjecture 253 must be taken into account. By order of magnitude, field energy density. For a static field this energy den- this sum can be estimated by placing all these Planck sity is negative with the negative mass density [29] masses at an average distance R, with R given by ( φ)2 ρ = − , (12.1) g π 2 = GM 4 Gc R 2 (11.19) c where φ is the scalar gravitational potential. and where M is the mass of the universe, equal to the With the inclusion of this negative mass density the sum of all Planck masses. Relation (11.19) follows Poisson equation becomes from general relativity, but it can also be justified if 1 2φ + ( φ)2 = 4πGρ. (12.2) one assumes that the total energy of the universe is c2 zero, with the positive rest mass energy Mc2 equal to the negative gravitational potential energy −GM 2/R. Making the substitution Summing up all contributions to dv/dt, one obtains 2 ψ = φ/c , from (11.18) and (11.19) e (12.3) 2 (12.2) is transformed into dv ≈ GM = c , 2 (11.20) 4πGρ dt R R 2ψ = ψ. (12.4) c2 as in Mach’s principle. Because the gravitational inter- In spherical coordinates with φ = c2 lnψ = 0 for r = action is solely determined by the gravitational mass, the equivalence of the inertial and gravitational mass is 0, (12.4) has for a sphere of constant density the solu- obvious. tion sinhkr 4πGρ According to Mach’s principle inertia results from ψ = , k2 = , (12.5) a “gravitational field” set up by an accelerated mo- kr c2 tion of all the masses in the universe relative to an ob- hence server. It is for this reason that the hypothetical grav- sinhkr itational field of Mach’s principle must act instanta- φ = c2 ln . (12.6) neously with an infinite speed. By contrast, the inertia kr in the Planck mass plasma is due to the presence of For small values of r one has the Planck masses filling all of space relative to which an absolute accelerated motion generates inertia as a φ =(2π/3)Gρr2 (12.7) purely local effect, not as a global effect as in Mach’s principle. Notwithstanding this difference in explain- with the force per unit mass ing the origin of inertia in the Planck mass plasma = − φ =( π/ ) ρ model and in Mach’s principle, the latter can be recov- g 4 3 G r (12.8) ered from the Planck mass plasma model provided not as in Newton’s theory. In general on has only all the masses in the universe are set into an accel- √ √ erated motion relative to an observer, but with them the 1 4πGρ 4πGρ g = c2 − ctnh r , (12.9) Planck mass plasma as well, or what is the same, the r c c physical vacuum of quantum field theory. Only then is complete kinematic equivalence established, and only which for r → ∞ yields the constant value then would Mach’s principle not require action at a g = − 4πGρc. (12.10) distance. This result implies the shielding of a large mass by 12. Gravitational Self-Shielding of Large Masses the negative mass of the gravitational field surrounding the mass. The shielding becomes important at the Einstein’s gravitational field equations are in the distance Planck mass plasma nonlinear field equations in flat c R = √ . (12.11) space-time, permitting to obtain an expression for the 4πGρ 254 F. Winterberg · Planck Mass Plasma Vacuum Conjecture Inserting into this expression the average mass density of magnitude smaller than the entropy given by (13.4). of the universe, the distance becomes about equal the This large discrepancy makes it difficult to under- radius of the universe as a closed curved space ob- stand the Bekenstein-Hawking black hole entropy in tained from general relativity. The great similarity of the framework of statistical mechanics. An expression the shielding effect by the negative mass of the gravita- for the entropy of a black hole, like the one given by tional field, with the electrostatic shielding of the pos- the Bekenstein-Hawking formula can be obtained from itive ions by the negative electrons in a plasma, there string theory, but in the absence of any other verifiable leading to the Debye shielding length, suggests that our prediction of string theory this should not be taken too universe is in reality a Debye-like cell, surrounded by seriously. The non-observation of Hawking radiation an infinite number of likewise cells, all of them sepa- from disintegrating mini-black holes left over from the rated from each other by the shielding length (12.11). early universe, rather should cast doubt on the very ex- In Einstein’s curved space-time interpretation of the istence of general relativistic black holes. General rel- gravitational field equations the field energy cannot be ativity tells us that inside the event horizon of a black localized. This is possible in the flat space-time inter- hole the time goes in a direction perpendicular to the pretation of the same equations. The negative value of direction of the time outside of the event horizon, a pre- this field energy implies negative masses, which is one diction which makes little sense and cannot be verified. of the two components of the Planck mass plasma. In the Planck mass plasma conjecture, Einstein’s field equations are true outside the event horizon for gravi- 13. Black Hole Entropy tational potentials corresponding to a velocity v < c.In approaching the event horizon where v = c, infalling Up to a numerical factor of the order unity, the matter acquires a large kinetic energy, and if the in- entropy of a black hole is given by the Bekenstein- falling matter is composed of Dirac spinor quasipar- Hawking [30, 31] formula ticles, these quasiparticles disintegrate in approaching the event horizon. Inside the event horizon Einstein’s = 3/ , S Akc Gh¯ (13.1) equation becomes invalid and physical reality is better where A is the area of the black hole (k Boltzmann con- described by nonrelativistic Newtonian mechanics. stant). As long as one is only interested in the order of With Einstein’s field equation valid outside the event magnitude for S, one can set horizon, Schwarzschild’s solution of Einstein’s gravitational field equation = 2, A R0 (13.2) dr2 ds2 = + r2 dθ 2 + sin2 θdφ 2 where 2Gm 1 − GM c2r (13.6) R = 0 2 (13.3) 2Gm c − c2 1 − dt2 c2r is the gravitational radius of a mass M. With the Planck length one can write for (13.1) still holds and can be given a simple interpretation in terms of special relativity and Newtonian gravity. Ac- S =(R /r )2k. (13.4) 0 p cording to Newtonian mechanics, a body falling in the 5 gravitational field of a mass M assumes a velocity v For a solar mass black hole, one has R0 ∼ 10 cm. With −33 given by rp ∼ 10 cm, the Bekenstein-Hawking entropy (in = ∼ 76 units in which k 1) gives S 10 . 2GM By comparison, the statistical mechanical expres- v2 = . (13.7) r sion for an assembly of N particles is (up to a logarithmic factor) given by In combination with special relativity, it leads to the length contraction S = Nk. (13.5) 57 2 Containing about ∼ 10 baryons, the statistical me- v 2GM dr = dr 1 − = dr 1 − (13.8) chanical entropy of the sun is S ∼ 1057, that is 19 orders c2 c2r F. Winterberg · Planck Mass Plasma Vacuum Conjecture 255 and the time dilation hence = / − 2/ 2 = / − / 2 . 2 2 dt dt 1 v c dt 1 2GM c r (13.9) m/mp = 1 − v /c = 1 − Rs/r (13.15) Inserting (13.8) and (13.9) into the line element 2 2 2 2 and therefore of special relativity ds = dr − c dt , one obtains 2 Schwarzschild’s line element (13.6). r − Rs ∼ m ≈ ∞ = . (13.16) Unlike dr and dt which are measured at r by an Rs mp outside observer far away from the mass M,dr and dt = are measured in the inertial system carried along the Inserting this into (13.13) and solving for t t 0, the infalling body. According to (13.7), the body reaches time needed to reach the distance r at which the ki- the velocity v = c at the Schwarzschild radius Rs = netic energy of an infalling particle becomes equal to 2 2GM/c = 2R0. The time needed to reach r = Rs is the Planck energy, one finds for the outside observer infinite, even though it is finite Rs mp 2 in the inertial system carried along the infalling body. t0 = ln a . (13.17) Since the same is true for a body as a whole collaps- c m ing under its own gravitational field, the gravitational After having disintegrated into Planck mass objects, collapse time for an outside observer is infinite. In the and reached the Schwarzschild radius at r = Rs where Planck mass plasma model, this time is finite because v ≈ c, the time needed to reach r = 0 is of the or- = in approaching r Rs the kinetic energy of all the ele- der Rs/c. The logarithmic factor therefore represents mentary particles, of which the body is composed, rises the increase of the gravitational collapse time over the / − 2/ 2 = / − / 2 in proportion to 1 1 v c 1 1 2GM c r, nonrelativistic collapse time Rs/c. Only in the limit until the energy of the elementary particles reaches the mp → ∞ does this time become infinitely long as in 22 Planck energy, at which the particles disintegrate. This general relativity. For an electron one has mp/m ∼ 10 happens at a radius somewhat larger than Rs. and hence In approaching the Schwarzschild radius R s, an in- m 2 falling particle assumes relativistic velocities which ln p =∼ 100, (13.18) permits to set in (13.6) ds = 0. The velocity seen by an m outside observer therefore is approximately given by making the collapse time about 100 times longer. In the example of a solar mass black hole one has Rs ∼ 1 km, dr Rs = −c 1 − (13.10) and hence for the nonrelativistic collapse time R /c ∼ dt r s 3×10−6 sec. The gravitational collapse time for a solar with the velocity of the infalling body measured in mass black hole, with the collapse starting from r = 2 its own inertial reference system about equal to c. 100Rs(a ∼ 10 ), would therefore be ∼ 100Rs/c ∼ 3 × From (13.10) one has 10−4 sec. During the gravitational collapse, the location of the − = dr ∼ dr ct = Rs (13.11) event horizon defined as the position where v = c, de- 1 − Rs/r r − Rs velops from ”inside out” [32]. Assuming a spherical and hence body of radius R and constant density ρ, the Newto- −ct/Rs nian potential inside and outside the body is r − Rs = const. e . (13.12) 2 If at t = 0, r =(a+1)Rs, a 1, it follows from (13.12) GM r φin = − 3 − , r < R; that 2R R (13.19) r − R − / s = ae ct Rs . (13.13) GM R φout = − , r > R. s r For an elementary particle of mass m to reach the An infalling test particle would at r = 0 reach the max- Planck energy and to disintegrate, it must have an ab- imum velocity v given by solute velocity equal to v2 3GM v/c = 1 − (m/m )2, (13.14) = −φin(0)= . (13.20) p 2 2R 256 F. Winterberg · Planck Mass Plasma Vacuum Conjecture Therefore, it would reach the velocity of light if the This increase is possible because the vacuum has an sphere has contracted to the radius unlimited supply of Planck mass particles in the superfluid groundstate. With the negative energy of the grav- 2 R1 = 3GM/c =(3/2)Rs = 3R0. (13.21) itational field, interpreted by an excess of negative over positive Planck mass objects, the gravitational collapse Accordingly, the disintegration of the matter inside the below r = R0 implies that a pair of positive and nega- sphere begins at its center after the sphere has col- tive Planck mass objects are generated out of the su- lapsed to (3/2) of its Schwarzschild radius. perfluid groundstate, with the positive Planck masses A qualitatively similar result is obtained from gen- accumulating in the collapsing body, and with the neg- eral relativity by taking the Schwarzschild interior so- ative Planck masses, from which the positive Planck lution for an incompressible fluid [33]. masses have been separated, surrounding the body. In passing through the event horizon (in going from The Planck mass objects, consisting of rotons etc., the subluminal to the superluminal state), the Planck cannot interact gravitationally for a distance of separa- mass plasma looses its superfluidity, and the vortex tion smaller than a Planck length. It is for this reason rings responsible for the formation of Dirac spinor that a spherical assembly of N Planck masses cannot quasiparticles disintegrate. What is left are rotons and collapse through the action of gravitational forces be- phonons, the quasiparticles in the Hartree approxima- low a radius less than tion. = 1/3 . To compute the black hole entropy, we make order r0 N rp (13.26) of magnitude estimates. If the negative gravitational The core of a solar mass black hole, for example, energy of a spherical body of mass M and radius R, would have a radius r ∼ 1019r ∼ 10−14 cm. Insert- 2/ 0 p which by order of magnitude is GM R, is set equal its ing (13.26) into (13.24), one has rest mass energy Mc2, the body has contracted to the 2 3/2 gravitational radius R0 = GM/c . After the body has N =(R0/rp) . (13.27) passed through the event horizon, its internal energy is confined and cannot be lost by radiation of otherwise. In statistical mechanics, the entropy of a gas composed In passing through the event horizon, the matter of the of N Planck mass particles mp with a temperature T is given by the Sackur- Tetrode formula [34] collapsing body has disintegrated into N0 Planck mass objects (like rotons), the number of which is estimated / / S = Nkln (V/N)(2πm kT)3 2e5 2/h3 , (13.28) by setting p / = 3 GM2 where V N rp, with the smallest phase space vol- N m c2 = = Mc2 (13.22) 3 3 3 0 p ume (∆p∆q) =(mpcrp) = h¯ . The temperature T in R0 the core of the black hole is the Davies-Unruh temper- and therefore ature [35,36] T = ha¯ /2πkc, (13.29) N0 = M/mp. (13.23) where a = GM/r2 is the gravitational acceleration at ∼ 33 ∼ −5 0 For a solar mass M 10 g (with mp 10 g), one r = r . With the help of (13.26) and (13.27) one obtains ∼ 38 0 has N0 10 . At the beginning of the collapse, one from (13.28) would have (in units in which k = 1) for the entropy 38 3/2 of the sun S ∼ N0 ∼ 10 . If the collapse proceeds to S =(5/2)(R0/rp) k =(5/2)Nk. (13.30) a radius r < R0, the number N of Planck mass objects increases according to The entropy of a black hole therefore goes here in proportion of the (3/4) power of the black hole area, in- GM2 stead in proportion of the area as in the Bekenstein- Nm c2 = (13.24) p r Hawking entropy. The entropy of two black holes with area A1 and A2 prior to their merger is here propor- or as tional to / = / . = 3/4 + 3/4, N N0 R0 r (13.25) S0 A1 A2 (13.31) F. Winterberg · Planck Mass Plasma Vacuum Conjecture 257 and after their merger proportional to Table 1. Hierachical displacement of Einstein and Yang- Mills field theories as related to the hierachical displace- 3/4 S1 = A , (13.32) ment of Newton point-particle, and Helmholtz line-vortex dynamics. where Newton point-particle Kine- Helmholtz line-vortex dynamics and Einstein’s matic dynamics and Yang-Mills / / / / 3/4 =( 3 4 + 3 4)3/2 > 3 4 + 3 4 gravitational field theory Quan-field theories A A1 A2 A1 A2 (13.33) tities as required by the second law of thermodynamics. r ψ velocity potential, f gauge functions With the black hole entropy proportional to the (3/4) Newtonian potential, φ r˙ −∇ψ Force on line power of its area, the entropy of a solar mass black metric tensor gik A vortex, gauge hole is reduced from the Bekenstein-Hawking value potentials S ∼ 1076 to S ∼ 1057, about equal the statistical me- Force on point particle, −∇φ r¨ W= Yang-Mills force gravitational force field Γ ∇ × A− field expressed N − chanical value if is set equal the number of baryons expressed by Christoffel g 1A × Aby charge space in the sun. symbols curvature tensor Einstein’s field equations R = 14. Analogies between General Relativity, expressed by metric- ×Γ +Γ ×Γ Non-Abelian Gauge Theories and Superfluid space curvature tensor Vortex Dynamics fields, which for a vanishing curvature tensor can al- In Einstein’s gravitational field theory the force on a ways be transformed away by a gauge transformation. particle is expressed by the Christoffel symbols which Likewise, if the curvature tensor in (14.2) vanishes, one are obtained from first order derivatives of the ten po- can globally transform away the gauge potentials. tentials of the gravitational field represented by the ten From the Newtonian point of view, contained in Ein- components of the metric tensor. From the Christoffel stein’s field equations, the force is the first derivative symbols the Riemann curvature tensor is structured by of a potential. Apart from the nonlinear term in (14.2) the symbolic equation this is also true for a Yang-Mills field theory. But with the inclusion of the nonlinear terms, (14.2) has also the R = Curl Γ + Γ× Γ. (14.1) structure of a curvature tensor. In Einstein’s theory the curvature tensor involves second order derivatives of The expression for the field strength, and hence force, the potentials, whereas in a Yang-Mills field theory the in Yang-Mills field theories is symbolically given by curvature tensor in charge space involves only first or- (g coupling constant with the dimension of electric der derivatives of the potentials. This demonstrates a charge) displacement of the hierarchy for the potentials with F = Curl A − g−1A × A. (14.2) regard to the forces. A displacement of hierarchies also occurs in fluid dynamics by comparing Newton’s It too has the form of a curvature tensor, albeit not in point particle dynamics with Helmholtz’s line vortex space-time, but in internal charge space, in QCD for dynamics. Whereas in Newton’s point particle dynam- example, in color space. It was Riemann who won- ics the equation of motion is mr¨ = F, the corresponding dered if in the small there might be a departure from equation in Helmholtz’s vortex dynamics is µr˙ = F, the Euclidean geometry. The Yang-Mills field theo- where µ is an effective mass [37]. Therefore, whereas ries have answered this question in a quite unexpected in Newtonian mechanics a body moves with constant way, not as a non-Euclidean structure in space-time but velocity in the absence of a force, it remains at rest in rather one in charge-space, making itself felt only in vortex dynamics. And whereas what is at rest remains the small. undetermined in Newtonian mechanics, it is fully de- Comparing (14.1) with (14.2) one can from a gauge- termined in vortex dynamics, where at rest means at field theoretic point of view consider the Christoffel rest with regard to the fluid. The same would have to Γ i symbols kl as gauge fields. If the curvature tensor van- be true with regard to the Planck mass plasma. ishes, they can be globally eliminated by a transforma- The hydrodynamics of the Planck mass plasma sug- tion to a pseudoeuclidean Minkowski space-time met- gests that the hierarchical displacement of the curva- Γ i ric. One may for this reason call the kl pure gauge ture tensor for Einstein and Yang-Mills fields is related 258 F. Winterberg · Planck Mass Plasma Vacuum Conjecture to the hierarchical displacement of the vortex equation 2 2 of motion and the Newtonian point particle equation of βH = 2µ ∑ ln|r j − rk| +(1/2 )∑|r j| , (15.2) motion. The hierarchical displacement and analogies ( jquarks cannot exist as free shift particles indicates that they are bound like phonons are bound in a solid. In accordance with this anal- ∆φ =(e/hc¯ ) A · ds =(e/hc¯ ) H · df (15.3) ogy, quarks may be the result of localized conden- sates of the Planck mass plasma and may only occur where H = curl A. Accordingly, there should be within these localized regions. This interpretation re- ceives support from condensed matter physics, where Z =(e/hc¯ ) H · df (15.4) fractional electron charges are observed in thin layers and are responsible for the anomalous quantum Hall effect. At least three points define a plane, making it vortices within the area df. To satisfy the Pauli prin- plausible why a minimum of three quarks are needed ciple there must be at least one vortex at the position to form a stable configuration. of each electron. In Laughlin’s wave function there are µ As Laughlin [39] has shown, an electron gas con- exactly vortices for each electron. We therefore have = µ fined within a thin sheet can be described by the wave to put Z . The fractional quantized Hall effect then / function simply means that the charge of one vortex is e 3 pro- µ = vided 3. It follows that in the two-dimensional µ −| |2/ 2 electron fluid each electron splits into three vortices of ψ(r ,r ,...r )= ∏ (z − z ) ∏e z j 4 , 1 2 N j k charge e/3. The quantization condition for the vortices ( jLorentz transformation formulas, and Sagnac’s have wave modes with a phase velocity exceeding the interpretation is not subject to any logical contradic- velocity of light. Because the nonlocal actions of quan- tion if the dynamic interpretation of special relativity tum mechanics cannot transmit any information, the is adopted. existence of superluminal phase velocities is all what is An effect where a phase shift on an electron wave needed. occurs in the absence of any electromagnetic forces has We will now show that with the Planck mass plasma been described by Aharonov and Bohm [46]. There, a conjecture, many of the mysteries surrounding quan- change in the magnetic vector potential alone can pro- tum mechanics don’t look so mysterious after all. If duce a shift. The effect is explained as a direct conse- physics has its root at the extremely high Planck en- quence of Schrödinger’s wave equation, into which the ergy, it should be of no surprise that effects projected potentials and not the force fields enter. For this reason from this energy scale down to the energy scale of ev- a case was made by Aharonov and Bohm to give the eryday life may look incomprehensible. potentials a more direct physical meaning rather than A good example for quantum mechanical nonlo- to be just a convenient mathematical tool. However, cality is the Aharonov-Bohm effect, and it is instruc- to elevate the potentials to true physical significance tive to compare it with the Sagnac effect. The out- has the problem that these potentials can always be come of his rotating interferometer experiment was changed by a gauge transformation without affecting used by Sagnac as a decisive argument against Ein- the physical results derived from them. Because the po- stein’s claim that physics could do without the aether tentials can measurably influence the phase of an elec- hypothesis [42]. Most text-books treating the Sagnac tron wave in the absence of any electromagnetic force effect, like the well-known theoretical physics course fields, it has alternatively been claimed that the effect by Sommerfeld [43], explain the effect by Sagnac’s is a proof for action at a distance in quantum mechan- original argument, that it is caused by a whirling mo- ics, where the magnetic force field inside a magnetic tion of the aether felt in a rotating reference system, but solenoid can influence the phase of the electron wave it is also often stated that the effect somehow is caused in the force-free region outside the solenoid. Instead by the centrifugal- and Coriolis-forces and for this rea- we will show that both the Sagnac and the Aharonov- son should be treated in the framework of general rela- Bohm effect can be understood to result from a rota- tivity. A discussion of the Sagnac effect within general tional aether motion, revealing a close relationship be- relativity can be found in the theoretical physics lec- tween these effects. This picture not only gives a full tures by Landau and Lifshitz [44], where it is alleged physical explanation of the potentials and the meaning that the effect is caused by the special stationary grav- of a gauge transformation but also eliminates the need itational field set up in a rotating reference frame, even for any hypothetical action at a distance. though there are no masses present as the source of a In the Sagnac effect, a light beam is split into two true gravitational field. Already Ives [45], had given separate beams, each one following a half circle of ra- convincing arguments that the effect has nothing to do dius r along the periphery of a rotating table. One of the with the inertial forces in a rotating reference frame, two beams is propagating in the same direction as the keeping Sagnac’s original argument to be valid as velocity of the rotating table, the other one in the oppo- ever. site direction. The rotating table shall have the angular The interpretation of the Sagnac effect, which says velocity −Ω, making its velocity at the radius r equal that it is caused by an aether wind in the rotating to u = −rΩ. reference system, would make in this system the ve- According to the hypothesis of an aether at rest locity of light anisotropic. Such a conclusion seems in the unaccelerated laboratory, the velocity of light to contradict the outcome of the Michelson-Morley judged from a co-rotating reference system would be and other experiments which claim to have proven c−v for the beam propagating in the same direction as the constancy of the velocity of light, which is why the rotating table and c + v for the beam propagating Einstein discarded the aether hypothesis. A close ex- in the opposite direction, where v = −u is the aether amination of all these experiments, however, shows velocity in the rotating reference system. The time dif- that one there never measures the one-way velocity ference for both beams leaving from the position A and 262 F. Winterberg · Planck Mass Plasma Vacuum Conjecture arriving at position B follows from The formula (10.22) can also be applied to a neutron interferometer placed on a rotating platform. An B B experiment of this kind, using the rotating earth as in cδt = (c + v)dt − (c − v)dt, (16.1) the Michelson-Gale version of the Sagnac experiment, A A was actually carried out [47], confirming the theoreti- cally predicted phase shift. hence Next we compute the phase shift (10.9) by a magnetic vector potential. To make a comparison with δ =( / ) . t 1 c vdt (16.2) the gravitational vector potential in the Sagnac effect, we consider the magnetic field produced by an in- If we are only interested in first order effects in v/c, finitely long cylindrical solenoid of radius R. Inside the ∼ we can put dt = (1/c)ds, where ds is the line element solenoid the field is constant, vanishing outside. If the along the circular path. If the light propagates along an magnetic field inside the solenoid is H, the vector po- arbitrary but simply connected curve we then have tential is (magnetic field directed downwards): 2 1 δt =(1/c ) v · ds, (16.3) Aϕ = − Hr , r < R, 2 (16.9) 1 HR2 for which we can also write = − , r > R. 2 r δ =( / 2) · . t 1 c curlv df (16.4) According to (10.9) the vector potential on a closed path leads to the phase shift where F = | df| is the surface enclosed by the light e path. Because Ω =(1/2)|curlv| we find δϕ = − Hπr2 , r < R, hc¯ δ = Ω / 2, e (16.10) t 2 F c (16.5) = − HπR2 , r > R. hc¯ which agrees with the result obtained by Sagnac. δϕ = ωδ ω As noted by Aharonov and Bohm [46], there is a phase For the phase shift t, where is the circu- > = > lar frequency of the wave, we have shift for r R, even though H 0 (because for r R, curlA = 0). A v δϕ =(ω/ 2) · . Expressing by (10.8) through , the hypothetical c v ds (16.6) circular aether velocity, one has e Alternatively, the phase shift can be seen to be caused vϕ = Hr , r < R, by the gravitational vector potential resulting from a 2mc (16.11) circular flow of the Planck mass plasma, with the prin- e HR2 = , r > R. ciple of equivalence precisely relating this circular flow 2mc r to the angular velocity of a rotating platform. Accord- ing to (10.17) one has for the gravitational vector po- One sees that inside the coil the velocity profile is the tential in a rotating frame of reference same as in a rotating frame of reference, having outside the coil the form of a potential vortex. If expressed in Aˆ = −vc = −Ωcr (16.7) terms of the aether velocity, the phase shift becomes m with the phase shift given by (10.22). For photons of δϕ = v · ds, (16.12) frequency ν, and mc2 = hν = h¯ω, one has h¯ the same as (10.22) for the vector potential created by a 2 2 2 δϕ =(ω/c ) v · ds = 2Ω(πr )(ω/c ), (16.8) gravitational field, and hence the same as in the Sagnac and the neutron interference experiments. For the mag- the same as predicted by Sagnac without quantum me- netic vector potential the aether velocity can easily be- chanics. come much larger than in any rotating platform experi- F. Winterberg · Planck Mass Plasma Vacuum Conjecture 263 ment. According to (16.11) the velocity reaches a max- entanglement in the many body Schrödinger equation imum at r = R, where it is in configuration space. In the Planck mass plasma this |v | eHR problem is avoided because it views all particles as max = . (16.13) quasiparticles of this plasma, and it is incorrect to vi- c 2mc2 sualize a many-body wave function to be composed of ∼ −4 For electrons this is |vmax|/c = 3 × 10 HR, where the same particles which are observed before an inter- H is measured in Gauss. Assuming that H = 104 G, action between the particles is turned on. In the pres- > > this would mean that |vmax| ∼ c for R ∼ 0.3 cm. It thus ence of an interaction it rather leads to a new set of seems to follow that the aether can reach superlumi- quasiparticles into which the wave function can be fac- nal velocities for rather modest magnetic fields. In this torized. This can be demonstrated for two identical par- regard it must be emphasized that in the Planck mass ticles moving in a harmonic oscillator well. The well plasma all relativistic effects are explained dynami- shall have its coordinate origin at x = 0, with the first cally, with the aether, which is here the Planck mass particle having the coordinate x1 and the second on the plasma, obeying an exactly nonrelativistic law of mo- coordinate x2. Considering two oscillator wave function. It can for this reason assume superluminal veloc- tions ψ0(x) and ψ1(x), with ψ0 having no and ψ1(x) ities, and if this would be the same velocity felt on a having one node, there are two two-particle wave func- rotating platform, it would lead to an enormous cen- tions trifugal and Coriolis-field inside the coil, obviously not observed. 2 −(x2+x2)/2 ψ(x ,x )=ψ (x )ψ (x )= x e 1 2 , The Planck mass plasma can give a simple expla- 1 2 0 1 1 2 π 2 nation for this paradox because it consists of two su- x perfluid components, one composed of positive Planck 2 masses and the other one of negative Planck masses. The two components can freely flow through each = (16.14a) x1 other making possible two configurations, one where both components are co-rotating, and one where they are counter-rotating. The co-rotating configuration is 2 −( 2+ 2)/ obviously realized on a rotating platform where it leads ψ( , )=ψ ( )ψ ( )= x1 x2 2, x1 x2 1 x1 0 x2 π x1e to the Sagnac and neutron interference effects. This suggests that in the presence of a magnetic vector x2 potential the two superfluid components are counter- rotating. Outside the coil, where curl A = 0, the mag- = ( . ) x 16 14b netic energy density vanishes, implying that the mag- 1 nitude of both velocities is exactly the same. Inside the coil, where curl A = 0, there must be a small imbalance in the velocity of the positive over the negative Planck graphically displayed in the x1, x2 configuration space, = = masses to result in a positive energy density. with the nodes along the lines x2 0 and x1 0. By Whereas in the Sagnac effect ω = curlv = 0, accord- a linear superposition of these wave functions we get a ing to (10.7) resulting in observable inertial forces, no symmetric and an antisymmetric combination: magnetic forces are present in the Aharonov-Bohm ef- 1 fect where curl A = 0. In the Sagnac effect the aether ψ (x ,x )=√ [ψ (x )ψ (x )+ψ (x )ψ (x )] s 1 2 0 1 1 2 1 1 0 2 makes a uniform rotational motion, whereas in the 2 Aharonov- Bohm effect the aether motion is a potential 1 −( 2+ 2)/ = √ (x + x )e x1 x2 2, vortex. With the kinetic energy of the positive Planck π 2 1 masses cancelled by the kinetic energy of the negative

Planck masses, the energy of the vortex is zero. Since x2 for shifting the phase no energy is needed, there can be = ( . ) no contradiction. x 16 15a An even more serious problem of quantum mechan- 1 ical nonlocality is the strange phenomenon of phase 264 F. Winterberg · Planck Mass Plasma Vacuum Conjecture 1 as quasiparticles of the Planck mass plasma, the ab- ψ (x1,x2)=√ [ψ0(x1)ψ1(x2) − ψ1(x1)ψ0(x2)] a 2 stract notion of configuration space and inseparability into parts disappears, because any many-body system 1 −(x2+x2)/ = √ (x − x )e 1 2 2, π 2 1 can, in principle, at each point always be expressed as a factorizable wave function of quasiparticles, where the quasiparticle configuration may change from point x2 to point. This can be shown quite generally. For an N- = ( . ) body system, the potential energy can in each point of x 16 15b 1 configuration space be expanded into a Taylor series

N U = ∑a x x . (16.19) If a perturbation is applied, whereby the two parti- ki k i k,i cles slightly attract each other, the degeneracy for the two wave-functions is removed, with the symmetric Together with the kinetic energy wave function leading to a lower energy eigenvalue. N For a repulsive force between the particles the reverse = mi 2. T ∑ x˙i (16.20) is true. As regards the wave functions (16.14), one may i 2 still think of it in terms of two particles, because the = + wave functions can be factorized, with the quantum po- one obtains the Hamilton function H T U, and tential becoming a sum of two independent terms: from there the many- body√ Schrödinger equation. In- troducing the variables m x = y , one has i i i 2 2 ψ∗ψ 2 ∂2 ψ∗ψ − h¯ = − h¯ 1 1 1 N N ∗ ∗ 1 2 ψ ψ ψ ψ ∂ 2 T = ∑ y˙ , U = ∑bkiykyi, (16.21) 2m 2m 1 1 x1 i (16.16) i 2 i 2 ∂2 ψ∗ψ − h¯ 1 2 2 . ψ∗ψ ∂ 2 which by a principal axis transformation of U becomes 2m 2 2 x2 N 1 N ω2 Such a decomposition into parts is not possible for the T = ∑ z˙2, U = ∑ i z2. (16.22) 2 i 2 i wave functions (16.15), and it is there then not more i i possible to think of the two particles which are placed Unlike the Schrödinger equation with the poten- into the well. This, however, is possible by making a ◦ tial (16.19), the Schrödinger equation with the poten- 45 rotation in configuration space. Putting tial (16.22) leads to a completely factorizable wave function, with a sum of quantum potentials each de- y = x + x , 2 1 pending only on one quasiparticle coordinate. The (16.17) x = x − x , transformation from (16.21) to (16.22) is used in clas- 2 1 sical mechanics to obtain the normal modes for a one obtains the factorized wave functions system of coupled oscillators. The quasiparticles into which the many-body wave function can be factorized 1 2 2 ψ = √ ye−(x +y )/2, are then simply the quantized normal modes of the cor- s π responding classical system. (16.18) 1 2 2 For the particular example of two particles placed ψ = √ xe−(x +y )/2, a π in a harmonic oscillator well, the normal modes of the classical mechanical system are those where the parti- for which the quantum potential separates into a sum cles either move in phase or out of phase by 180 ◦.In of two independent terms, one depending only on x quantum mechanics, the first mode corresponds to the and the other one only on y. This means that the addi- symmetric, the second one to the antisymmetric wave tion of a small perturbation, in form of an attraction or function. It is clear that the quasiparticles representing repulsion between the two particles, transforms them the symmetric and antisymmetric mode cannot be lo- into a new set of two quasiparticles, different from the calized at the position of the particles placed into the original particles. With the identification of all particles well. F. Winterberg · Planck Mass Plasma Vacuum Conjecture 265 The decomposition of a many-body wave function into a factorizable set of quasiparticles can in the course of an interaction continuously change, but it can also abruptly change, if the interaction is strong, in what is known as the collapse of the wave function. 17. Planck Mass Rotons as Cold Dark Matter and Quintessence With greatly improved observational techniques a number of important facts about the physical content and large scale structure of our universe has emerged. They are: 1. About 70% of the material content of the universe is a negative pressure energy, 2. About 26% nonbaryonic cold dark matter, Fig. 4. The phonon-roton energy spectrum of the hypothet- 3. About 4% ordinary matter and radiation, ical Planck aether. 4. The universe is Euclidean flat, ter energy. The roton hypothesis can therefore explain 5. It’s cosmological constant very small, both the cold dark matter and negative pressure energy, the latter mimicking a cosmological constant [49]. 6. It’s expansion slightly accelerated. 18. Conclusion These are the basic facts which have to be explained, and no model which at least can make them plausible Finally, I try to compare the presented model with can be considered credible. other attempts to formulate an unified model of ele- In the Planck mass plasma model with an equal mentary particles. number of positive and negative Planck mass parti- I begin with Heisenberg’s nonlinear spinor theory cles the cosmological constant is zero and the uni- [50]. As with Einstein’s nonlinear gravitational field verse Euclidean flat. In its groundstate the Planck mass theory, it is massless and selfcoupled, and like Ein- plasma is a two component positive-negative mass su- stein’s gravitational field it resists quantization by es- perfluid with a phonon-roton energy spectrum for each tablished rules. In his theory Heisenberg tries to over- component. Assuming that the phonon-roton spectrum come this problem by a novel kind of regularization. measured in superfluid helium is universal, this would Because this leads to a Hilbert space with an indef- mean that in the Planck mass plasma this spectrum inite metric, a high prize has to be paid. To obtain has the same shape, with the Planck energy taking the solutions describing elementary particles Heisenberg place of the Debye energy in superfluid helium (see uses perturbation theory, which contradicts the spirit of Fig. 4), with the roton mass close to the Planck mass. the entire theory, because perturbation theory should Rotons can be viewed as small vortex rings with the come into play only after a spectrum of elementary ring radius of the same order as the vortex core radius. particles had been obtained non-perturbatively. Pertur- A fluid with cavitons is in a state of negative pressure, bation theory would have to describe the interactions and the same is true for a fluid with vortex rings [48]. between the nonperturbatively obtained particles. As In vortices the centrifugal force creates a vacuum in noted by Heisenberg, one fundamental problem of the the vortex core, making a vortex ring to behave like a theory is that there should be at least one fermion with caviton. a nonzero rest mass, absent from his nonlinear spinor The kinetic roton energy is bound by the height of wave equation. the potential well in frequency space. From Fig. 4 it By comparing Heisenberg’s theory with the Planck follows that the ratio of the energy gap (which is equal mass plasma vacuum model, the following can be said: the roton rest mass energy), to the maximum kinetic ro- For a linear classical field without interaction, quanti- ton energy is about 70 to 25, close to the observed ratio zation is like a postscript, obtained by quantizing the of the negative pressure energy to the cold dark mat- normal modes of the classical field with Planck’s har- 266 F. Winterberg · Planck Mass Plasma Vacuum Conjecture monic oszillator quantization rule. The fundamental be formulated as a nonlinear field theory in flat space- field equation of the Planck mass plasma is nonlinear time. but nonrelativistic. By the Madelung transformation it Comparing string theory with the Planck mass assumes the form of an Euler equation. As in classi- plasma model, one is wondering if closed strings are cal fluid dynamics it is nonrelativistic and nonlinear. misunderstood quantized vortex rings at the Planck The fundamental solutions of this Euler equation are scale, and if supersymmetry is mimicked by the waves and vortices and they can be readily quantized. hidden existence of negative masses. According to And with elementary particles understood as quasipar- Schrödinger, Hönl, and Bopp, Dirac spinors are ex- ticle configurations made up from the waves and vor- plained as being composed of positive and negative tices, perturbation theory can there be done in the usual masses, dynamically explaining supersymmetry. In the way. The prize to be paid is to give up the kinematic Planck mass plasma the problem of quantum gravity is space-time symmetry of Einstein’s special theory of reduced to the quantization of a non-relativistic many relativity, replacing it by the dynamic pre-Einstein the- body problem. ory of relativity of Lorentz and Poincaré, which still Most recently a different attempt to formulate a fun- assumed the existence of an aether, here identified with damental theory of elementary particles has been made the Planck mass plasma. by ‘tHooft [53]. As in the Planck mass plasma model Presently, the most popular attempt to formulate a it is assumed that at the Planck scale there are two unified theory of elementary particles, and to solve the types of particles, called “beables” and “changeables.” problem of quantum gravity, is supersymmetric string The “beables” are massless non-interacting fermions. theory in 10 space-time dimensions, or in its latest ver- Because they form a complete set of observables that sion 11 dimensional M-theory. But M-theory, and by commute at all times, the “beables” are deterministic implication string theory, stands or falls with super- and can be described by an infinite sheet moving in symmetry. There are two main features which speak a direction perpendicular to this sheet with the veloc- in favor of supersymmetry: ity of light. The quantum fuzziness enters through the First, supersymmetric theories are less divergent be- “changeables,” interacting with the “beables.” The the- cause often, but not always, divergent Feynman dia- ory is thus a classical-quantum hybrid. grams cancel each other out. Second, supersymmetry Comparing ‘tHooft’s theory with the Planck mass makes the strong, the weak and the electromagnetic plasma, where fermions are composed of positive and coupling constants to converge at very high energies negative masses, the “Zitterbewegung” length (which almost into one point. Without supersymmetry the con- is 1/2 the Compton wave length of a fermion) di- vergence is much less perfect. verges with a vanishing rest mass, making understand- What speaks against supersymmetry is the follow- able why the “beables” are extended over an infinite ing: First, the supersymmetric extension of the stan- sheet. A further difference between t’Hooft’s theory dard model leads to electric dipole moments of the and the Planck mass plasma is that ‘t’Hooft’s theory electron and neutron which are not observed. Super- sustains the kinematic interpretation of the special the- symmetric string theories lead to even larger dipole ory of relativity, and by implication of the general the- moments [51]. Second, in spite of enormous efforts ory of relativity, with the existence of black holes in- made, no supersymmetric particles have ever been side event horizons, while in the Planck mass plasma, found, neither in particle accelerators up to F ∼ where Lorentz invasiance is a dynamic symmetry, ele- 200 GeV (tevatron), nor in the cosmic radiation at mentary particles disintegrate in approaching the event much higher energies. horizon, leading to the formation of non-relativistic red As an extension of Einstein’s general theory of rel- holes. ativity to higher dimensions, string theory leads to A theory in 3 space and 1 time dimension, inspired all the pathological solutions of general relativity, like by condensed matter physics, appears to be much more the Gödel-type travel back in time solution, and even plausible than a string theory with 6 more space dimen- worse, to the “NUT” solution but Newman, Unti and sions, M-Theory with 7 more space dimensions or F- Tamborini [52]. These solutions result from the topo- Theory with 7 more space- and one more time dimen- logical oddities of a Riemannian curved space. sion, and also more plausible than the hybrid classical- As Heisenberg had told me, because of the topo- quantum theory of “beables” and “changeables.” My logical problems in Einstein’s theory, gravity should theory is based on the simple conjecture that the vac- F. Winterberg · Planck Mass Plasma Vacuum Conjecture 267 uum of space is a kind of plasma of locally interacting Bopp, whose work was stimulated by the work of positive and negative Planck masses, leading to quan- Schrödinger. Because all these authors had published tum mechanics and the theory of relativity as low en- in German, their work did not get the attention it de- ergy approximations. served. Finally, I would like to express my sincere thanks to Acknowledgement my colleagues H. Stumpf and H. Dehnen, for their crit- One purpose of this report is to remind the physics ical reading of my manuscript and their many valuable community of the almost forgotten work by Hönl and comments.

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