aa r X i v : . [ phy s i c s . g e n - ph ] M a r Energy and Mass Generation
B.G. SidharthInternational Institute for Applicable Mathematics & Information SciencesHyderabad (India) & Udine (Italy)B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India)
Abstract
Modifications in the energy momentum dispersion laws due to anoncommutative geometry, have been considered in recent years. Weexamine the oscillations of extended objects in this perspective andfind that there is now a ”generation” of energy.
Many modern approaches, particularly in Quantum Gravity introduce a min-imum spacetime scale. It is well known that the introduction of such afundamental minimum length in the universe leads to a noncommutativegeometry. On the other hand there has been much discussion about howthis could avert the infinities which plague Quantum Field Theory (Cf.ref.[1]and several other references therein). As pointed out by Snyder many yearsago [2], there are now new commutation relations which replace the usualQuantum Mechanical relations. These are[ x, y ] = ( ıl / ¯ h ) L z , [ t, x ] = ( ıl / ¯ hc ) M x , etc. [ x, p x ] = ı ¯ h [1 + ( l/ ¯ h ) p x ]; (1)where a is the fundamental length.It was noted by the author that these new commutation relations lead to amodified energy momentum relation viz., [1, 3, 4] E = p + m + αl p (2)1iving the so called Snyder-Sidharth Hamiltonian [5], α being a scalar, and l the fundamental length, in units c = 1 = ¯ h . This leads to a modification ofthe usual Klein-Gordon equation, which now becomes,( D + l ∇ − m ) ψ = 0 (3)where D denotes the usual D’Alembertian.It is well known that the usual Klein-Gordon equation, without the extraterm in (2) describes a super position of normal mode harmonic oscillators[6]Let us first start with the Klein-Gordon equation itself( D + m ) ψ ( x ) = 0 (4)Following a well known procedure, a Fourier integral over elementary planewaves is then taken to give ψ ( x, t ) = Z d k q (2 π ) ωk h a ( k ) e ık.x − ıω k t + a † ( k ) e − ık.x + ıω k t i = Z d k h a ( k ) f k ( x ) + a † ( k ) f ∗ k ( x ) i (5)where ω k = + √ k + m and f k ( x ) = 1 q (2 π ) ω k e − ık.x An inversion then gives Z f ∗ k ( x, t ) ψ ( x, t ) d x = 12 ω k h a ( k ) + a † ( − k ) e ıω k t iZ f ∗ k ( x, t ) ˙ ψ ( x, t ) d x = − ı h a ( k ) − a † ( − k ) e ıω k t i (6)Finally the canonical Quantum Mechanical commutation relations for thewave function (now operator) φ and the associated momentum leads to(Cf.ref.[6] for details)[ a ( k ) , a † ( k ′ )] = Z d xd y [ f ∗ k ( x, t ) ∂ ψ ( x, t ) , f k ′ ( y, t ) ∂ ψ ( y, t )]= + ı Z d xf ∗ k ( x, t ) ∂ f k ′ ( x, t ) = δ ( k − k ′ )2imilarly [ a ( k ) , a ( k ′ )] = ( ı ) Z d xd y [ f ∗ k ( x, t ) ∂ ψ ( x, t ) , f ∗ k ′ ( x, t ) = 0and [ a † ( k ) , a † ( k ′ )] = 0 (7)This finally gives the Hamiltonian in terms of the creation and annihilationoperators a and a † viz., H = 1 / Z d kω k [ a † ( k ) a ( k ) + a ( k ) a † ( k )] (8)Discretizing the above, replacing the integrals in momentum space by sum-matiions and the Dirac-Delta function by the Kronecker Delta, we get, as iswell known, H = X k H k = X k / ω k ( a † k a k + a k a † k ) a k = q ∆ V k a ( k ) (9)with [ a k , a † k ′ ] = δ kk ′ [ a k , a k ′ ] = [ a † k , a † k ′ ] = 0 (10)In this second quantized picture, these lead to the various particle solutionsgiven by H k Φ k ( n k ) = ω k ( n k + 1 / k ( n k ) (11) Let us see, how all this gets modified due to the new considerations and inparticular the new commutation relations (1). We first observe that if insteadof the usual energy momentum relation, we use (2) above, then this wouldlead to a modification of the frequency given in (5). That is we would have ω k = √ k + m + αl k ≡ ω ′ k (12)(Remembering that we are in natural units). This would mean that therewould be a shift in the energy of the individual oscillators (Cf. also [7, 8]).But also, there are other important changes.3o see this, let us consider the above in greater detail. As is well known forthe individual Quantum Harmonic oscillators in (9) or (11) we have [6] a = λ q + ıλ pa † = λ q − ıλ p (13)where λ = (cid:16) mω (cid:17) and λ = (cid:16) mω (cid:17) , q replaces x and we have dropped thesubscript k for ω . This leads to, in the old case the Hamiltonian H = ω ( a † a + 12 ) (14)In our case however there is the additional term in (2). To see this in greaterdetail we note that owing to the commutation relations (1) we have[ q, p ] = ı (1 + l p ) (15)and so, [ a, a † ] = (1 + l p ) (16)Whence we have H = mω aa † + a † a ]= mω a † a + (1 + l p )]= mω [ a † a + 12 (1 + l p )] (17)Finally we have H = mω [( a † a + 12 ) + l p ] (18)= H ′ + mω l p where H ′ is given by H ′ ≡ mω ( a † a + 12 ) (19)that is, it would be the Hamiltonian if l were 0.Using (13) it is easy to see that p is given by p = ( mω a − a † ) (20)4sing (20) in (18), we get finally H = H ′ + l H mω − ( mωl ( a + a † ) (21)Equation (21) for the Quantum Mechanical Harmonic Oscillator alreadyshows the effect of the extra term in the SS-Hamiltonian (2). This is anadditional energy that appears due to the commutation relations (1).But it also shows that apart from the shift in energy (or mass) for the individ-ual oscillators, reflecting (2), the eigen states of the oscillators get mixed updue to the presence of the squares of the creation and annihilation operators.The eigen states would be combinations of states like Φ( n − , Φ( n + 2) inaddition to Φ( n ), of the usual theory as in (11). In other words we have toconsider a system of oscillators. This is due to the fact that a point is beingreplaced by an extension.Indeed in String Theory itself, as we know we have a similar situation [1] and[9]-[12]. We have ρ ¨ y − T y ′′ = 0 , (22) ω = π l s Tρ , (23) T = mc l ; ρ = ml , (24) q T /ρ = c, (25) T being the tension of the string, l its length and ρ the line density and ω in (23) the frequency. The identification (23),(24) gives (25), where c isthe velocity of light, and (22) then goes over to the usual d’Alembertian ormassless Klein-Gordon equation and an expansion in normal modes (whichare uncoupled).Further, if the above string is quantized canonically, we get h ∆ x i ∼ l . (26)Thus the string can be considered as an infinite collection of harmonic oscil-lators [10]. Further we can see, using equations (23) and (24) and the factthat ¯ hω = mc and that the extension l is of the order of the Compton wavelength in (26),a circumstance that was called one of the miracles of the string theory by5eneziano [13, 14, 15, 16, 17].The point is that a non zero extension for the system of oscillators wouldimply the commutation relations (1). This again implies an extra energyor mass on the one hand, and a coupled system of oscillators on the other,rather than a superposition of normal modes as we would otherwise have.Thus a consideration of a system of coherent oscillators would be a betterstarting point, if we consider the Planck scale, as in string theory and otherapproaches, rather than spacetime points of the usual theory. This circum-stance has been overlooked.In summary there is now an extra energy or mass that appears due to (1)and (2), on the one hand. On the other, we have to consider a system of(coupled) oscillators. We know that String Theory, Loop Quantum Gravity, the author’s own ap-proach of Planck oscillators, and a few other approaches start from the Planckscale.We will consider the problem from a different point of view, which also en-ables an elegant extension to the case of the entire Universe itself [18, 19,20, 21]. Let us consider an array of N particles, spaced a distance ∆ x apart,which behave like oscillators that are connected by springs. This is becauseof the results in the last section that we have to consider coupled oscillators.The starting point for such a programme can be found in [22].We then have[24, 25, 26, 19] r = √ N ∆ x (27) ka ≡ k ∆ x = 12 k B T (28)where k B is the Boltzmann constant, T the temperature, r the extent and k is the spring constant given by ω = km (29) ω = km a ! r = ω ar (30)6e now identify the particles with Planck masses and set ∆ x ≡ a = l P ,the Planck length. It may be immediately observed that use of (29) and(28) gives k B T ∼ m P c , which of course agrees with the temperature of ablack hole of Planck mass. Indeed, Rosen [27] had shown that a Planck massparticle at the Planck scale can be considered to be a Universe in itself aSchwarzchild Black Hole of radius equalling the Planck length. We also usethe fact deduced earlier to that a typical elementary particle like the pioncan be considered to be the result of n ∼ Planck masses.Using this in (27), we get r ∼ l , the pion Compton wavelength as required.Whence the pion mass is given by m = m P / √ n Further, in this latter case, using (27) and the fact that N = n ∼ , and(28),i.e. k B T = kl /N and (29) and (30), we get for a pion, rememberingthat m P /n = m , k B T = m c l ¯ h = mc , which of course is the well known formula for the Hagedorn temperature forelementary particles like pions [28]. In other words, this confirms the earlierconclusions that we can treat an elementary particle as a series of some 10 Planck mass oscillators.However it must be observed from (30) and (29), that while the Planck massgives the highest energy state, an elementary particle like the pion is in thelowest energy state. This explains why we encounter elementary particles,rather than Planck mass particles in nature. Infact as already noted [20],a Planck mass particle decays via the Bekenstein radiation within a Plancktime ∼ − secs . On the other hand, the lifetime of an elementary particlewould be very much higher.In any case the efficacy of our above oscillator model can be seen by thefact that we recover correctly the masses and Compton scales in the order ofmagnitude sense and also get the correct Bekenstein and Hagedorn formulasas seen above, and further we even get the correct estimate of the mass andsize of the Universe itself, as will be seen below.Using the fact that the Universe consists of N ∼ elementary particleslike the pions, the question is, can we think of the Universe as a collection of nN or 10 Planck mass oscillators? This is what we will now show. Infactif we use equation (27) with ¯ N ∼ ,
7e can see that the extent is given by r ∼ cms which is of the order ofthe diameter of the Universe itself. We shall shortly justify the value for ¯ N .Next using (30) we get¯ hω ( min )0 h l P i − ≈ m P c × ≈ M c (31)which gives the correct mass M , of the Universe which in contrast to theearlier pion case, is the highest energy state while the Planck oscillatorsindividually are this time the lowest in this description. In other words theUniverse itself can be considered to be described in terms of normal modesof Planck scale oscillators.More generally, if an arbitrary mass M , as in (31), is given in terms of ¯ N Planck oscillators, in the above model, then we have from (31) and (27): M = √ ¯ N m P and R = √ ¯ N l P , where R is the radius of the object. Using the fact that l P is the Schwarzchildradius of the mass m P , this gives immediately, R = 2 GM/c a relation that has been deduced alternatively. In other words, such anobject, the Universe included as a special case, shows up as a Black Hole, asuper massive Black Hole in this case. The interesting question is, from where does the extra energy come? Wemust remember that in the above considerations we have an a priori darkenergy background, and it is this energy that manifests itself as the extraenergy or mass or frequency.This can be illustrated by considering the background electromagnetic field,or the Zero Point Field as a collection of ground state oscillators (Cf.ref.[1]and references therein). It is known that the probability amplitude is ψ ( x ) = (cid:18) mωπ ¯ h (cid:19) / e − ( mω/ h ) x x from its position of classical equilibrium.So the oscillator fluctuates over an interval∆ x ∼ (¯ h/mω ) / The background electromagnetic field is an infinite collection of independentoscillators, with amplitudes X , X etc. The probability for the various os-cillators to have amplitudes X , X and so on is the product of individualoscillator amplitudes: ψ ( X , X , · · · ) = exp [ − ( X + X + · · · )]wherein there would be a suitable normalization factor. This expression givesthe probability amplitude ψ for a configuration B ( x, y, z ) of the magnetic fieldthat is described by the Fourier coefficients X , X , · · · or directly in termsof the magnetic field configuration itself by, as we saw, ψ ( B ( x, y, z )) = P exp − Z Z B ( x ) · B ( x )16 π ¯ hcr d x d x ! .P being a normalization factor. At this stage, we are thinking in terms of en-ergy without differenciation, that is, without considering Electromagnetismor Gravitation etc as separate. Let us consider a configuration where themagnetic field is everywhere zero except in a region of dimension l , where itis of the order of ∼ ∆ B . The probability amplitude for this configurationwould be proportional to exp[ − ((∆ B ) l / ¯ hc )]So the energy of fluctuation in a region of length l is given by finally, thedensity [29, 19] B ∼ ¯ hcl So the energy content in a region of volume l is given by β ∼ ¯ hc/l (32)Equation (32) can be written asEnergy ∼ m e c ∼ ¯ hcl (33)9quation (33) shows that l is the Compton wavelength of an elementaryparticle, which thus arises naturally. On the other hand if in (33), we considerthe energy of a Planck mass rather than that of an elementary particle, wewill get the Planck length.For another perspective, it is interesting to derive a model based on the theoryof phonons which are quanta of sound waves in a macroscopic body [30].Phonons are a mathematical analogue of the quanta of the electromagneticfield, which are the photons, that emerge when this field is expressed as a sumof Harmonic oscillators. This situation is carried over to the theory of solidswhich are made up of atoms that are arranged in a crystal lattice and canbe approximated by a sum of Harmonic oscillators representing the normalmodes of lattice oscillations. In this theory, as is well known the phononshave a maximum frequency ω m which is given by ω m = c π v ! / (34)In (34) c represents the velocity of sound in the specific case of photons, while v = V /N , where V denotes the volume and N the number of atoms. In thismodel we write l ≡ (cid:18) πv (cid:19) / l being the inter particle distance. Thus (34) now becomes ω m = c/l (35)Let us now liberate the above analysis from the immediate scenario of atomsat lattice points and quantized sound waves due to the Harmonic oscillationsand look upon it as a general set of Harmonic oscillators as above. Then wecan see that (35) and (32) are identical as ω = mc ¯ h (36)Using (36), we can once again recover both the Planck length and an ele-mentary particle Compton wavelength. On the other hand it has been shownthat starting with the background Zero Point Field the Quantum Mechanicalcommutation relations in J XJ yield the Quantum Mechanical spin at theCompton wavelength [1]. It is to be noted that this Quantum Mechanical10pin feature is absent at the Planck length.We finally comment the following. If in the considerations of Section 2 wetake the particle to have negligible mass or vanishing mass as in the case ofradiation, then this new physics as embodied in (18) leads to an increased fre-quency of the radiation, or an apparent increase in its velocity leading to dif-ferent wavelengths travelling with different speeds. The effect is very minuteand can be observed only for very high frequency radiation like Gamma Raysfrom Gamma Ray Bursts. Already there are claims that such lags in arrivaltimes of the Gamma Rays have indeed been found [31].
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