fulltext

Anomalous red-shifts due to a Dirac-like temporal variation

of a discrete-time interval in quantum theory

C. WOLF

Department of Physics, North Adams State College - North Adams, MA 01247, USA

(ricevuto il 17 Giugno 1996; approvato il 23 Luglio 1996)

Summary. — By considering a Dirac-like time variation of the discrete-time interval in discrete-time quantum theory we demonstrate that any anomalies in the red-shift of spectral lines can be used to place limits on the Dirac-like temporal evolution of the fundamental discrete-time interval (t) in the theory.

PACS 98.80 – Cosmology.

1. – Introduction

In the past two decades there has been sufficient interest in the theoretical community in trying to understand the pregeometric origin of space-time in terms of more primitive and fundamental notions. The motivations for such interest come from two directions, firstly because both quantum gravity and quantum field theory in general have failed to be consistent theories, at least at very small distances investigators have sought to study both gravitational theory and field theory on a finite discrete lattice [1-4], secondly because of the paradoxical and ambiguous interpretations of quantum theory various students of the early universe have sought to re-investigate the arena (space-time) of quantum theory in terms of more fundamental discrete notions [5, 6]. Among the investigators in this direction, Wheeler [7, 8] has sought to replace the continuum by a discrete sequence of fundamental yes-no choices with space-time and quantum theory emerging after a combinatoric sequence of these choices. Finkelstein [9] has discussed the idea of a quantum discrete net with the continuum being an average over the “quantum net”, and Wootters [10] has discussed the generic origin of space-time in terms of quantum correlations between fundamental spins in the Hilbert space. Other investigators who have erected discrete pregeometric origins of space-time in terms of graph theory, combinatorics, discrete sets and topology include Evako [11], Antonsen [12], Kull et al. [13], and Bombelli et al. [14]. The basic theme of all these studies is that after a discrete model of space-time and quantum field theory has been constructed, by allowing the discrete space and time interval to become small, both Minkowski space and conventional field theory emerge.

A different approach to discreteness was suggested long ago by Caldirola [15, 16], namely due to the fact that Minkowski space-time has an uncertain origin, fluctuations from this continuum of Minkowski space-time should appear in the quantum equation of motion. Recami [17] has interpreted this as a fundamental uncertainty principle in time where each particle has its own sense of time and only after thermal averaging, quantum correlations and statistical mixing was Minkowski space-time born. However, when an external Hamiltonian acts on a particle’s wave function, the wave function should respond over a finite time interval about t (Minkowski time) to reflect this uncertainty between the “particle’s time” and the averaged-out “time” of Minkowski space. Thus finite differences should appear in the quantum equations of motion. Actually Heisenberg had this idea quite long ago but applied it to the spatial variable [18]. We have applied the idea of discrete-time differences in quantum theory to electron spin polarization precession [19], electron spin resonance [20], spectral shifts in the hydrogen [21], and to various other problems involving the internal properties of elementary particles [22-25].

In the present paper we point out that the discrete-time interval (t) which is due to the uncertain tuning between the particle’s frame and the frame of synchronous observers (averaged-out frame) might vary with cosmological time scales [26] and in fact we would expect fluctuations in the early universe to be large away from any average values because of the possibility of non-trivial topology and space-time foam created by quantum-gravitational effects [27]. In view of this, if a spectral line is emitted at an earlier cosmological time and if the spectral line is sensitive to discrete-time effects, we would expect the red-shift of this spectral line to depend not only on the Hubble constant and deceleration parameter but also it depends on the discrete-time interval at earlier cosmic times. When we take these effects into account, we arrive at a formula for the red-shift that has the usual parameters (H , q0) in it, but

also the red-shift now has extra terms dependent on the wavelength which is a specific feature of discrete-time quantum mechanics and the variation of t over cosmological time scales. This is the purpose of this paper, namely to demonstrate that a specific signature in the red-shift results due to cosmological-time variation of the discrete-time interval (t).

2. – Red-shift due to cosmological-time variation of the discrete-time intervaln

We begin by writing down the formula for the frequency of absorption for an atomic transition in discrete-time equation theory as [21] (t, discrete-time interval)

v 4 2 tsin 21

g

EFt 2 ˇ

h

2 2 tsin 21

g

EIt 2 ˇ

h

(2.1)

(EF, EI4 final and initial energies), EFD EIand in the emission of a photon we have the

same frequency. After we expand eq. (2.1) we find to order t2

v C EF2 EI ˇ 1 t2 24 ˇ3(E 3 F2 E 3 I) 4v01 t 2 A

or f 4f01 t2_A 2 p 4 f01 t 2_{A .} (2.2)

Here f04 normal frequency shift of spectral line without t present,

t2_{A 4} t 2 2 p

g

1 24 ˇ3

h

(E 3 F2 EI3) 4

4 correction term due to discrete-time quantum mechanics . If t varies with cosmological evolution due to the fact that pregeometric fluctuations become damped, we might expect a relationship tt 4const, or tR(t) 4const

(

where R(t) 4cosmological scale factor

)

for the variation of the discrete-time interval. If we choose tR 4const, we have tR(t) 4t0R(t0) where t 4time spectral line is emitted, t0

time spectral line is received. For the Robertson-Walker line element we have for the propagation of a photon [28] in the radial direction

( dS)2 4 dt22 R 2 dr2 1 2Kr2 4 0 (K 40, 61) (2.3) giving (here C 41) dt R(t) 4 2 dr ( 1 2Kr2)1 O2 or



t t0 dt R(t) 4 2



r1 0 dr ( 1 2Kr2₎1 O2 4



0 r1 dr ( 1 2Kr2₎1 O2

for the emission and reception of a wave crest, also for a wave crest one wavelength later we have



t 1dt t01 dt0 dt R 4



0 r1 dr ( 1 2Kr2₎1 O2 , thus



t0 t01 dt0 dt R(t) 1



t t0 dt R(t) 2



t t 1dt dt R(t) 4



t t0 dt R(t) giving dt0OR(t0) 4dtOR(t), since (reinserting C) l04 C dt0, we have

l0

l 4

R(t0)

R(t) (2.4)

tR(t) 4t0R(t0): l04 l R(t0) R(t) 4 R(t0) C R(t) f 4 CR(t0) R(t)

g

f01 t2 0R2(t0) A R2_(t)

h

(2.5)

(

after using t0R(t0) 4tR(t)

)

. We now expand

R(t) 4R(t0) 1R . (t0)(t 2t0) 1R O (t0) (t 2t0)2 2 ! and rewrite eq. (2.5) to order t20 and (t 2t0)2 after using

R2_{(t) 4R}2(t0) 12R(t0) R . (t0)(t 2t0) 1 RO(t0) R(t0) 2 (t 2t0) 2 1 R .₂ (t0)(t 2t0) 2 and f01 t20 R2(t0) R2_(t) A 4f

y

1 1 t2 0 f A

u

22 R . R R2 (t 2t0) 2 ROR 2 R2(t 2t0) 2 2R .₂ (t 2t0)2 R2 1 4 R2_R.2 R4 (t 2t0) 2

vz

(here f 4f01 t20A 4frequency that would be emitted at t0 or also the frequency

apparently emitted if we did not take into account the Dirac variation of t),

(2.6) l04

u

1 1 R. R(t02 t) 1

u

R. R

v

2 (t02 t)2

u

1 2 ROR 2 R.3

v

v

C f Q Q

u

1 1 t 2 0A f

u

22 R R . R2 (t02 t) 1 ROR 2 R2(t02 t) 2 23

u

R . R

v

2 (t02 t)2

v

v

.

Here R, R., RO_{refer to values at t 4t}0. In eq. (2.6) we identify R

.

OR 4 H 4 Hubble const, q04 2R

O ROR.2

4 deceleration parameter, and f 4 COlAI (lAI4 wavelength apparently

emitted or the wavelength that would be emitted at t0when t 4t0). The red-shift now

reads from eq. (2.6)

(2.7) _{Z 4} l0 lAI 2 1 4 H(t02 t) 1 H2(t02 t)2

g

1 1 q0 2

h

1 1lAIt 2 0A C

C

`

D

22 H(t02 t) 1 ROR 2 R2(t02 t) 2 25 H2(t02 t)2

E

`

F

.

We see from eq. (2.7) that we get the normal red-shift which is wavelength independent and a second term which diminishes the red-shift and is wavelength dependent.

3. – Conclusion

The first two terms in eq. (2.7) represent the usual red-shift formula where the second term applies to red-shifts over large cosmological spatial separations. The terms proportional to lAI represent a wavelength-dependent red-shift which would

provide us with clear signals of both discrete-time effects and the variation of t over cosmological time scales. We also note that the determination of H0 from the

Tully-Fisher method gives a value of H0of 80 km per second per megaparsec and the

determination of H0 from type-I Supernova gives a value of 50 km per second per

megaparsec [29]. The margin of error in each does not, however, exceed the difference of the two values. When looking for wavelength-dependent shifts a specific value of H0

should be adopted. We also note that there are different red-shift diagrams for different portions of the spectrum [30], for instance, the optical diagram of Z vs. distance has a tendency to curve up at red-shifts from 1 to 2, while the infrared diagram remains a straight line for Z between 1 and 2. The best way to test eq. (2.7) is to look at the radio galaxies where l and Z are large, if characteristic lines can be identified in the radio spectrum and the red-shift of a series of these lines decreases with increasing l in proportion to l, then this would be evidence that the wavelength- dependent term in eq. (2.7) is a manifestation of discrete-time effects and the cosmological variation of t. We note that all of the terms that are multiplied by lAIin eq. (2.7) give a decreasing

red-shift. Thus a plot of red-shift vs. lAIat a given cosmological distance would provide

us with an experimental test of eq. (2.7), a diminishing of Z with increasing l would signal discrete-time effects and the cosmological variation of t. The coefficient in eq. (2.7)

(

(lAIt20A) OC

)

for radio waves of l B1 Met gives

( 1 )(t20) C

g

E ˇ

h

3 B t 2 0 108( 10 8₎3 B 10163 t20.

Even for t0C 10218s [21], the coefficient would give a very small correction and it

would be difficult to distinguish it from statistical fluctuations and local motions of the galaxies. However, if a specific radio galaxy yielded a slow decrease in Z vs. l, it would provide us with evidence for the presence of the wavelength-dependent shift in eq. (2.7).

* * *

The author would like to thank the Physics Departments at Williams College and Harvard University for the use of their facilities.

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