Locus model for space-time fabric and quantum indeterminacies
aa r X i v : . [ g r- q c ] M a y Locus model for space-time fabric and quantumindeterminacies
Alberto C. de la Torre ∗ Universidad Nacional de Mar del PlataArgentina
A simple locus model for the space-time fabric is presented and is compared withquantum foam and random walk models. The induced indeterminacies in momentumare calculated and it is shown that these space-time fabric indeterminacies are, inmost cases, negligible compared with the quantum mechanical indeterminacies. Thisresult restricts the possibilities of an experimental observation of the space-timefabric.Keywords: atomic space-time, Planck scale, indeterminacies.
I. INTRODUCTION
In this work a very simple atomic or discrete space-time model is presented. Theatoms of space-time, that we name loci ( locus in singular), have sizes comparablewith Planck scale and are located in a mathematical continuous space. All pointswithin a given locus are physically equivalent and can not be differentiated or takenapart.We can consider these loci as probability distributions for physical coordinatesand therefore distances, time intervals and momenta become random variables thatcan be calculated from the loci distributions. We will find the probability uncer-tainty for these quantities, without any reference to quantum mechanics, and wewill afterwards compare them with the corresponding quantum mechanic indeter-minacies. II. THE LOCUS MODEL
Let us use the continuous real variables ( x, t ) to denote the mathematical coor-dinates of a space-time point. We will differentiate these mathematical coordinates ∗ Electronic address: [email protected] from the physical coordinates because two points x and x separated by a distancecomparable with the Planck length ℓ p are physically indistinguishable and two in-stants of time separated by an interval comparable with Planck time t p can not beconsidered as physically different. In order to formalize this concept we proposethat a physical coordinate is described by space-time region, a locus , centered at amathematical coordinate and having a width given by the Planck scale. A space-time localization of a particle means that a certain locus is occupied and a physicalspace-time interval will be determined by the set of loci between two space-timepoints. The precise shape and boundaries of these loci will not be relevant and wecan also imagine soft boundaries that could be described by a probability density(gaussian for instance). In this way, physical coordinates become random variablesdistributed with probability densities L x,ℓ p ( ξ ) and T t,t p ( θ ) with center at ( x, t ) andwidths ( ℓ p , t p ). The physically relevant interval, or distance between two coordinates x and x , is then a random variable x − x distributed according to the convolution Z ∞∞ dη L x ,ℓ p ( ξ − η ) L x ,ℓ p ( η ) . (1)Let us consider now the physical space between two distant coordinate points, say x = 0 and x = ℓ . We can define a partition of the interval 0 < x < x , · · · , < x N < ℓ and we have ℓ = ℓ − x N + x N − x N − + x N − − · · · + x − x + x −
0. So we havedecomposed the interval ℓ in a sum of N + 1 subintervals and therefore this physicallength is a random variable distributed as an N + 1 fold convolution. If N is large,the distribution will approach a gaussian distribution, regardless of the shape of the locus distribution, with a width given by √ N − ℓ p . Accordingly, the distributionof a physical length ℓ will depend on the number of points in the partitions. Wecan fix the number of subintervals to be approximatively equal to the number of loci fitting in the length ℓ . Let us define then δ L to be the space density of loci andthen we have N = δ L ℓ . We can expect that the space density of loci is close to 1 /ℓ p because if the density would be much larger, then we could have physical locationsseparated by a distance less then ℓ p and if it were much smaller the transition fromone location to the next would not be possible. A physical length ℓ , much longerthan Planck length ℓ p , is then a random variable with a gaussian distributionΞ( ξ ) = 1 √ πσ x exp (cid:18) − ( ξ − ℓ ) σ x (cid:19) (2)peaked at ξ = ℓ and with a width σ x (we reserve ∆ x for quantum indeterminacies) σ x = p ℓδ L ℓ p , (3)and if we take δ L ≈ /ℓ p we have σ x = (cid:18) ℓℓ p (cid:19) / ℓ p . (4)In a similar way we conclude that a time interval t , much longer than Plancktime t p , is a random variable with a gaussian distributionΘ( θ ) = 1 √ πσ t exp (cid:18) − ( θ − t ) σ t (cid:19) (5)with a width σ t σ t = p tδ T t p , (6)where δ T is the loci time density and if we take it δ T ≈ /t p we have σ t = (cid:18) tt p (cid:19) / t p . (7)We can now compare these results with other models with an essential indeter-minacy in space-time points. An early proposal was made by Karolyhazy[1] thatcombined Heisenberg’s uncertainty principle with Schwarzschild horizon in order toestimate a minimal uncertainty in the measurement of a distance ℓ and a time t given by σ x ∼ (cid:18) ℓℓ p (cid:19) / ℓ p , and σ t ∼ (cid:18) tt p (cid:19) / t p . (8)Due to the 1 / locus model. These results, obtained from a heuristic argument,were rediscovered in the context of a quantum foam model for the space-time fabric[2]and they gain support from other apparently independent arguments[3]. Indeed itwas shown that the same result can be obtained as a consequence of the HolographicPrinciple and also from Black Holes physics and even from Information and Com-puter Theory. The locus model result, with an 1 / / III. INDUCED INDETERMINACIES IN MOMENTUM
In this section we will deduce the momentum indeterminacy induced by the space-time indeterminacies. Although we will concentrate on the indeterminacies of the locus model, the conclusions are also valid for the other models with a 1 / m moving a distance ℓ during a time interval t . Since these quantities are random variables with distributions given in Eqs.(2, 5)the momentum of the particle, given by p = mℓ/t , will also be a random variablewith the distribution corresponding to the quotient of random variables. Thereforethe momentum p will be distributed according to the probability density functionΠ( ̟ ) given byΠ( ̟ ) = Z ∞−∞ dξ Z ∞−∞ dθ Ξ( ξ ) Θ( θ ) δ (cid:18) ̟ − m ξθ (cid:19) = Z ∞−∞ dθ (cid:12)(cid:12)(cid:12)(cid:12) θm (cid:12)(cid:12)(cid:12)(cid:12) Ξ( ̟θ ) Θ( θ ) . (9)If we insert the gaussian densities given in Eqs.(2,5) we can obtain the momentumdensity distribution in terms of the Error Function and considering that t ≫ t p weget the approximationΠ( ̟ ) ≈ √ pmc π ̟ + mc ( ̟ + pmc ) / (cid:18) ℓℓ p (cid:19) / exp − ( ̟ − p ) pmc ℓ p ℓ ( ̟ + pmc ) ! . (10)This distribution is peaked at ̟ = p = mℓ/t and its width can be estimated from thedenominator of the exponent. However we can obtain the momentum indeterminacymore rigorously from the definition σ p = Z ∞−∞ d̟ ( ̟ − p ) Π( ̟ )= Z ∞−∞ d̟ Z ∞−∞ dξ Z ∞−∞ dθ ( ̟ − p ) Ξ( ξ ) Θ( θ ) δ (cid:18) ̟ − m ξθ (cid:19) = Z ∞−∞ dξ Z ∞−∞ dθ ( m ξθ − p ) Ξ( ξ ) Θ( θ )= m Z ∞−∞ dξ ξ Ξ( ξ ) Z ∞−∞ dθ θ Θ( θ ) − mp Z ∞−∞ dξ ξ Ξ( ξ ) Z ∞−∞ dθ θ Θ( θ ) + p = m ( σ x + ℓ ) (cid:28) θ (cid:29) − mpℓ (cid:28) θ (cid:29) + p = m σ x (cid:28) θ (cid:29) + m ℓ (cid:18)(cid:28) θ (cid:29) − t (cid:28) θ (cid:29) + 1 t (cid:19) . (11)Since the Θ( θ ) distribution is sharply peaked at θ = t we can take as good approxi-mation h /θ i = 1 /t and h /θ i = 1 /t and with this, the parenthesis in last equationvanishes. With this we become then the simple expression σ p p = σ x ℓ . (12)This is a general result but if we specialize it for the locus model, using Eqs.(4 and7), we obtain σ p p = σ x ℓ = σ t t (cid:18) mcp (cid:19) / = (cid:18) ℓ p ℓ (cid:19) / . (13) IV. COMPARISON WITH QUANTUM INDETERMINACIES
In the estimation of the indeterminacies due to the space-time fabric with a 1 / / / σ x and σ p with the quantum mechanical indeterminacies ∆ x and ∆ p , inparticular with their correlations ∆ x ∆ p ≥ ~ /
2. From Eq.(13) above we immediatelyobtain for the product of the space-time fabric indeterminacies σ x σ p = 2 pℓ p , (14)and therefore for a momentum smaller than some value we have a product of indeter-minacies below the quantum mechanical bound ~ /
2. This value of momentum turnsout to be enormous, 5 kg m/s or 10 ev/c, many orders of magnitude bigger thanthe highest energy cosmic rays observed. We must conclude that the indeterminaciesdue to the space-time fabric lie deep below the quantum mechanical indetermina-cies. This places severe limits on the observability of the space-time indeterminaciesin the kinematics of a particle because any observation will encounter the quantummechanical limits long before the space-time indeterminacies are approached. Thereare proposals[8] to observe space-time indeterminacies in extragalactic light by inter-ferometric techniques taking ℓ large enough, close to the observable universe length ℓ ∼ m. In such a long travel, the photons in the locus or random walk modelswould develop an indeterminacy σ x ∼ . − m, much longer than the wave length,making the light incoherent and therefore no interference should be observed. In thecase of the other models, the indeterminacy accumulated is σ x ∼ − m, not suf-ficient to destroy coherence. The observation of interference fringes seem to excludethe 1 / / σ x ∼ − m the quantum mechanical indeter-minacy must be even smaller ∆ x < σ x but this implies a momentum indeterminacy∆ p > ~ / (2 σ x ) and this turns out to be 30 Mev/c requiring very high energy gammarays. V. CONCLUSIONS