L]~TTERE AL NUOVO CIMENTO YOL. 40, x. 13 28 Luglio 1984

Does Wave-Particle Duality Apply to Galaxies?

~[. DERSARKISSIAN

Department o/ Physics, Temple University - Philadelphia, P A (19122), U S A .

(ricevuto il 22 Dicembre 1983)

PACS. 03.65. - Quantum theory; quantum mechanics. PACS. 04.60. - Quantum theory of ~ 'avi tat ional .

Summary . - The recent discovery of quantized red-shifts for galaxies suggests a cosmic form of quantum theory which may parallel tbc form so familiar on the atomic- distance scale or smaller.

When DE BROGLIE postulated wave-particle duality for all matter, his focus was on the atomic-distance scale or smaller. Quantum effects are only evident on this scale, because Planck's constant (h) is extremely small: ~ 10 2/erg-s. Every student of quantum mechanics knows how de Broglie's idea was developed into a formal theory to describe the quantum behaviour of atomic matter. The results are well-known: uncertainty principles for incompatible observables; the need for noncommuting al- gebra; the probability interpretation of the wave function; etc.

It now appears in the realm of possibility that wave-particle duali ty may apply on the galactic distance scale, with (h) replaced by a much larger value (hg), in order to make quantum effects visible. This possibility has emerged with the fascinating recent discovery that galactic red-shifts appear to be quantized (~), with a high confidence level. However, it should be carefully noted that the high confidence level refers to the existence of the red-shift quantization effect. The confidence level appears to be lower for answers to three important questions:

l) Is the effect strictly limited to the internal dynamics associatcd with galactic structure, or is it also present for the red-shift arising from the motion of point galaxies (relative to a ((galactic observer ~): i.e. an observer located at the centre-of-mass of the malky way galaxy)? The data reported by TIFFT in ref. (1) appear to deal with the former. However, a reasonable extrapolation of Tifft's idea suggests that red- shift quantization may exist for both cases (possibly, with the same velocity increment, Av, for adjacent red-shift states).

(1) ~V. TIFFT: Astrophys. J., 221, 756 (1978).

390

DOES WAYE-PAKTICLE DUALITY APPLY TO GALAXI]~S ~. 391

2) Wha t is the velocity increment (Av) for adjacent red-shift s tates (henceforth referred to as the <~ Tifft interval rule ~>)? The l imited da ta presently available tenta- t ive ly suggests two Tifft interval rules (with the dist inct possibil i ty of var ia t ion on the following themes): Av = 72 km/s and/or Av is equal to an appropria te sub- multiple of 72 km/s (probably 12 km/s).

3) Is Av = constant in the Tifft in terval rule(s)~ Constancy implies tha t the red-shift s tate of a dis tant point galaxy (relative to a galactic observer) is given by the rule: AVg = N(Av), where N is an appropria te integer and AVg is related to the red- shift parameter in the s tandard, nonrelativist ic way: AZ = (AVg)/v, where c is the speed of l ight. A t the present time, the Tifft interval rule appears to be known only in relation to galactic structure and the value Av ---- constant ---- 12 km/s is favoured.

Except for the above finite structure, which should be resolved through the interac- t ion between theoret ical models and experimental data, nature appears to be suggesting tha t the well-known difficulties associated with measuring the position and the l inear momentum of a galaxy simultaneously may simply reflect the incompat ib i l i ty of these observables (in the sense of ordinary quantum mechanics). In this case, the existence of a cosmic uncer ta in ty principle and an associated cosmic quantum mechanics seems quite plausible. Ordinary quantum mechanics then serves as an essential guide to the development of the new theory. These unorthodox ideas are briefly explored in this paper ; an expanded research program is briefly summarized at the end.

This prel iminary investigation excludes consideration of the much more difficult problem of constructing a cosmic quantum model to explain the red-shift quantization effect in a single galaxy with structure; in a double galaxy (both galaxies with or without structure); ... etc. Consider a point galaxy (mass = ~g) subject to the Hubble expansion law. I t is possible to construct a l inear momentum operator for the point ga laxy by noting two impor tant properties of galaxies:

1) The ga laxy has a <~cosmic, l inear momentum due to Hubble expansion: pc = (ca r Ho) x, where H 0 = the Hubble parameter evaluated at t = t~ = the present cosmic t ime and the position vector from a galactic observer to the galaxy is denoted by x. The cosmic l inear momentum is thought to be due to the effects of ear ly universe dynamics.

2) The ga laxy has a (~local~) l inear momentum: PL ~ Pw + P x , where p~ is the component of p~ parallel to po and p • is the component perpendicular to po. The local l inear momentum is thought to be due to the local gravi ta t ional environment through which the galaxy moves. A reasonable assumption is tha t the local gravi ta t ional environment largely determined the size, shape and rotat ion s tate of the galaxy.

The to ta l l inear momentum of the ga laxy is given by

(1) p = p c + p ~ .

The red-shift of interest arises from the component of p in the z-direction; call i t p~. Now use ordinary quantum mechanics as a guide in constructing the following Hermi t ian operator for the l inear momentum of the point galaxy:

d (2) p~ = + (m~ H0) ~ - - ih~ d x '

where h~ = h J 2 z and h~ is the cosmic Planck 's constant. From cq. (2), construct the

3 9 2 3~. D]~RSAKISSIAN

complex conjugate operator

(3) d

Given ~ ~* (p~, p~), i t is a s t ra ightforward ma t t e r to construct the Hermi t ian operator, /~(, defined as follows:

(4) f< ~ ~ [ (~x)2 2 . 2 , , ~ (P*)'~] ] "

The operator, /~, is a lmtural candidate for the radial kinet ic-energy operator for the point galaxy moving wi th nonrela t iv is t ic velocity. After some algebra, eq. (4) can be pu t in the form

(5) R - - - -h~ d 2 + 1 2~*~ dx 2 2 (m:H~ " "~

~uppose the q u a n t u m state of the point galaxy is represented by the wave func- t ion, ~J(x). The wave fm~ction squared ( = I~,i s} conta ins informat ion about the posit ion and l inear m o m e n t u m (i.e., the red-shift) of the galaxy as i t appeared a t its (,look- back t ime )) (the t ime when radia t ion was emi t ted from a younger galaxy to be received on E a r t h at the present cosmic t ime is equal to to). Now use ord inary q u a n t u m mechanics as a guide to in te rpre t I~,1 ~ as a probabi l i ty densi ty and to construct the eigenvalue equa t ion for the operator, f<:

(6) R,j, = ~-,p.

S i n c e / ~ is Hermi t ian , its e igenvalues (e) are real and its e igenfunet ions Of) form a eoln- plete, o r thonormal set.

The solution of eq. (6), us ing eq. (5) for /~, is formally ident ical to the eigenvalue problem for the nonrelat ivis t ie , l inear harmonic oscillator (2). Therefore,

1) the eigenwdues are quant ized:

(7) s,~ = h~ (,)(,~ + ,~),

where ~o : H~ and (n) = 0, 1, 2 . . . . , co;

2) the e igenfunet ion have the form

(S) ~,~(x) = An exp [-- ~2/2] H,df ix) ,

where A n - a normal iz ing cons tant -- %/f i / (2"n!~ /~ i ,

= (fi) x,

Hn(fix ) = the Hermi te polynomial of order (~t).

The q u a n t u m number , (n), defines the allowed red-shift eigenstates for point galaxies; [Fnl '2 gives the one-dimensional spatial d is t r ibut ion funct ion.

(~) Pt. WINTER: Quantum Mechanics (Wadsworth Publishing Co., 1979), p. 80.

DOES WAVE-PARTICLE DUALITY APPLY TO GALAXIES? 393

The spacing between adjacent red-shift eigenstates is determined from the condition

(9) e~+ 1 - e~ = (/i~ H0)/2 .

Equation (9), after some algebra, predicts the following Tifft interval rule for point galaxies:

m.,(Av)= g . ( H ~ (lO)

Equation (10) has the expected 4 classical l imit )> (i.e., ]~--+ 0 implies that Av--+ 0); it also shows that Av is not constant for point galaxies, but decreases lowly with n--approaching zero like 1/(~,)+. This result appears to be physically reasonable, as the following remarks demonstrate. Observation of galactic red-shifts for large x (i.e., large n) means observation of galaxies at an early stage in their evolntion. The galaxies were closer together then and had larger mutual g~'avitational effects. Local and cosmic velocities were larger than they are now (for <( nearby >> galaxies) and red- shift states then were packed closer together than they are now: i.e. Av then was smaller than it is now (for <( nearby ~> galaxies).

Equation (10) can be used to estimate (h~). With n = 0, the result is

(11) h~ ~ (1 + V~) ~ %(Av)2 H0

With m: = (10) 44 g, H 0 = 50 km/s/Mpc and Av ~ 12 km/s, eq. ( l l ) gives the numerical result

(12) hg ~ 7" 10 ~4 (erg)(s) .

The commutator of (4,15x) follows from eq. (2):

(13) [4,p~] = i ~ .

From a general theorem (a) in ordinary quantum mechanics, which requires only Schwartz's inequali ty and the representation of observables by Hermitian operators for its proof, the uncertainty principle for the incompatible observables (x, p~), which have an operator representation satisfying eq. (13), has the form

(14) (• ~> ~:/2,

where (Apx , Ax) are defined in the root mean square sense. Equation (14) may be interpreted in the following way: the linear momentum of a galaxy is uncertain by an amount Ap, = mg(Av,), where Av, is approximated by ]%1, which has a typical value: (250--300) km/s. Then Ax should be comparable to the diameter of the galaxy. A rough calculation gives Ax = (10) 5 light years, which is roughly the diameter of the disk in the spiral galaxy Andromeda (which has v~ = - 270 km/s).

('~) R. ~VINTER: Quantum Mechanics (Wadsworth Publishing Co., 1979), p. 114.

3 ~ ~[. D:ERSAKISSIAN

I t appears tha t the cosmic form of the uncer ta inty principle is giving reasonable results and tha t galactic quantum mechanics is, indeed, plausible. However, there are impor tant topics tha t need to be explored. They will serve as lhe basis for the extended research program now in prog~ress. The list includes

1) the three-dimensional extension of this research,

2) the relat ivist ic extension of this research,

3) the problem of galactic structure and its relation to the Tifft interval rule and angular-momentum quant izat ion,

4) the connection between (h~) and other fundamental constants of nature ,

5) the possible role of the visible universe in defining boundary conditions on the galactic wave function,

6) inclusion of the red-shift quantization effect in Einstein 's theory of general re la t iv i ty ,

7) the cosmic analog of the SchrSdinger wave equation (for the nonrelativist ie problem) and the Klein-Gordon and Dirac equations (for the relat ivist ic problem).

These topics (and others) make this new research field both fertile and fascinating.

I an indebted to Dr. J. CROW, who (as Chairman of the Physics Department) made i t possible for me to teach quantum mechanics at the same t ime I was exploring prob- lems in cosmology. The ideas presented in this paper began to cristallize during tha t period. The ideas were developed with par t ia l support from a Facu l ty ]~esearch Fellowship, awarded to the author by Temple Universi ty for Summer, 1983.