Particle production and transplanckian problem on the non-commutative plane
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Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
PARTICLE PRODUCTION AND TRANSPLANCKIAN PROBLEMON THE NON-COMMUTATIVE PLANE
MASSIMILIANO RINALDI
D´epartment de Physique Th´eorique, Universit´e de Gen`eve,24 quai Ernest Ansermet CH–1211 Gen`eve 4, [email protected]
We consider the coherent state approach to non-commutativity, and we derive from it aneffective quantum scalar field theory. We show how the non-commutativity can be takenin account by a suitable modification of the Klein-Gordon product, and of the equal-timecommutation relations. We prove that, in curved space, the Bogolubov coefficients areunchanged, hence the number density of the produced particle is the same as for thecommutative case. What changes though is the associated energy density, and this offersa simple solution to the transplanckian problem.
Keywords : Minimal length, QFT on curved space, transplanckian problem04.60.Bc, 04.62.+v
1. Introduction
In recent years, we have seen many proposals aimed to quantize consistently thegravitational field from fundamental principles. From a phenomenological point ofview, a more modest approach consists in introducing reasonable modifications toquantum field theory and look for observable consequences in black holes or in-flationary models. For example, one can construct a theory where the dispersionrelations depart from linearity above a certain energy scale, thus breaking localLorentz invariance. This approach is motivated by analogue models of gravity incondensed matter systems 1, by deformations of the Lorentz group 2, or by tensor-vector models of gravity 3. Modified dispersion relations were considered in thecontext of renormalization, and particle production in curved spacetime 4 , , ,
7. Inthe latter case, the common result is that the thermal spectrum seen from an accel-erated detector or from an asymptotic observer on a black hole background is onlymarginally affected by non-linear dispersion relations 8 , , ovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Massimiliano Rinaldi
In contrast to these proposals, here we would like to study how quantum fieldtheory is modified when spacetime has an intrinsic non-commutative structure.This topic has been intensively investigated by assuming that non-commutativeeffects in field theory are implemented by replacing the ordinary product amongfunctions with the so-called star-product 13 , , , ,
17. Instead, in this paper wewould like to take onboard an alternative point of view, and consider the coherentstate approach to non-commutativity introduced in a series of papers by E. Spallucciand collaborators 18 , , ,
21. As we will briefly explain below, this model doesnot need the star-product, since all non-commutative effects are encoded in theGaussian damping of the field modes a . As a result, the field propagator is finite inthe ultraviolet limit, but the dispersion relation is the same as the relativistic one.Compared to the wider class of modified theories mentioned above, this proposalhas a stronger predictive power, as both field modes and propagators are known.For example, this model has been already studies in connection with the Casimireffect 24, and inspired several works on black holes 25 , , , , , , , , ,
2. Non-commutative field theory
To begin with, let ˆ z , ˆ z be the coordinate operators of a two-dimensional non-commutative plane, that satisfy the algebra 18[ˆ z , ˆ z ] = iθ . (1)Generalizations to higher even-dimensional spaces are straightforward 19. One candefine the new operators ˆ A = ˆ z + i ˆ z , ˆ A † = ˆ z − i ˆ z , (2)which satisfy the canonical commutator [ ˆ A, ˆ A † ] = 2 θ . The coherent states associatedto these operators are the states | α i such that ˆ A | α i = α | α i . Their explicit form reads | α i = exp (cid:20) θ (cid:16) α ∗ ˆ A − α ˆ A † (cid:17)(cid:21) | i , (3)and one can show that h α | α i = 1. Physical, commuting coordinates are c-numbers,defined as the expectation values on coherent states of the coordinate operators, a On mathematical grounds, in this theory the star-product can be seen as replaced by the so-calledVoros product 22 , ovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Particle production and transplanckian problem on the non-commutative plane namely x ≡ h α | ˆ z | α i = Re( α ) , x ≡ h α | ˆ z | α i = Im( α ) . (4)The vector ( x , x ) is interpreted as the mean position of the point particle on thenon-commutative plane. So, how does non-commutativity show up in quantum fieldtheory? The answer lies essentially in the fact that, now, every function F ( ~x ) mustbe promoted to an operator ˆ F (ˆ z , ˆ z ) evaluated on coherent states. In particular,this holds for the monochromatic wave function, which is turned into the operatorexp( i~p · ˆ ~z ), where ~p = ( p , p ) is a real vector. After having defined the transversemomenta p ± = ( p ± ip ) /
2, one can utilize the Baker-Campbell-Hausdorff formulato find b h α | e ip ˆ z + ip ˆ z | α i = h α | e i p + ˆ A † e i p − ˆ A | α i e p + p − [ A † ,A ] = e i~p · ~x − θ ( p + p ) . (5)We see that the main effect of the intrinsic non-commutative structure of spaceresides in the damping term embedded in the plane wave operator. We will showlater that this term can also be interpreted as a deformation of the measure in theFourier integral. There is one crucial aspect in the expression (5): the relative signin the quadratic term ( p + p ) is independent of the signature of the metric on theplane. It simply arises from the product p − p + , hence its form is the same in bothEuclidean and Lorentzian signatures.Now, let us consider a scalar field φ ( t, x ) with mass m , which propagates on atwo-dimensional Minkowski space, and is governed by the standard Klein-Gordonequation c ( (cid:3) + m ) φ ( t, x ) = 0 (6)According to the prescriptions above, the positive frequency field modes are modifiedaccording to u p ( t, x ) = e − ℓ ( ω + p ) √ πω e − iωt + i~p · ~x . (7)In this expression, we identify p with ω = p p + m , ℓ = θ/
4, and we set p = p .As the modes are solutions to Eq. (6), we define the corresponding Klein-Gordonproduct 38, which, however, must be modified to accommodate to the differentnormalization of the modes. Thus, we have a “damped” δ -function( u p , u p ′ ) = − i Z dx ( u p ←→ ∂ t u ∗ p ′ ) = e − ℓ ( ω + p ) δ ( p − p ′ ) , (8) b We stress that the components of ~p are c-numbers, so they act trivially on the coherent states.There is a more formal approach in terms of non-commutative Fock space, but the results are thesame 22. c In our notation, ds = dt − dx . ovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Massimiliano Rinaldi and this reflects the fact that coherent states are normalized but not orthogonal.The scalar field can be represented as the usual mode sum φ ( t, x ) = Z d~p √ πω (cid:2) ˆ a p u p ( t, x ) + ˆ a † p u ∗ p ( t, x ) (cid:3) , (9)where ˆ a p is the annihilation operator, which fulfills the standard rule [ˆ a p , ˆ a † p ′ ] =4 πωδ ( p − p ′ ). It follows that the equal-time commutator reads[ φ ( t, x ) , ˙ φ ( t, x ′ )] = i √ πℓ e − ℓ m − ( x − x ′ ) / ℓ . (10)In the vanishing ℓ limit, the right-hand side smoothly tends to iδ ( x − x ′ ). TheWightman’s positive frequency function can be determined in the usual way, andthe result is G + ( x µ , x ′ µ ) ≡ h | φ ( x µ ) φ ( x ′ µ ) | i = Z d~p πω e − ℓ ( ω + p ) − ip µ ( x µ − x ′ µ ) , (11)from which it follows that the Feynman propagator reads G F ( x µ , x ′ µ ) = − i Z d~p πω e − ℓ ( ω + p ) h θ ( t − t ′ ) e − ip µ ( x µ − x ′ µ ) + θ ( t ′ − t ) e ip µ ( x µ − x ′ µ ) i . (12)This expression can be derived from the integral (for p = ω and p = p ) G F ( x µ , x ′ µ ) = i Z d p (2 π ) e − ℓ ( p + p ) − ip µ ( x µ − x ′ µ ) p − p − m , (13)from which we can easily read off the momentum space propagator˜ G F ( p , p ) = e − ℓ ( p + p ) p − p − m , (14)which is the same found via path integral quantization 18. It is easy to check thatit satisfies the equation( (cid:3) + m ) G F ( x µ , x ′ µ ) = − i πℓ e − ( ∆ t +∆ x ) / ℓ , (15)where ∆ t = ( t − t ′ ) and ∆ x = ( x − x ′ ) . The right-hand side becomes the usual δ -function for ℓ → G EF ( p , p ) = e − ℓ ( p + p ) p + p + m = e ℓ m Z ∞ ℓ ds e − s ( p + p + m ) . (16)The corresponding Euclidean propagator in coordinate space is G EF ( x µ , x ′ µ ) = e ℓ m Z ∞ ℓ ds Z ∞ dp dp π e − ip x − ip x − s ( p + p + m ) . (17)By integrating, we find the following cases:ovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Particle production and transplanckian problem on the non-commutative plane • for ℓ = 0 we have the usual massive two-dimensional scalar propagator,which shows a logarithmic divergence as m → x + ∆ x ) → • for ℓ = 0 and m = 0, the integral does not converge for s → ∞ . However,one can integrate between ℓ and some arbitrary L > ℓ , and then ex-pand for (∆ x + ∆ x ) →
0. The leading term of the series is constant andproportional to ln(
K/L ), thus it is irrelevant when quantities obtained bydifferentiating the propagator, such as the stress tensor, are computed. Wenote that this situation is peculiar to the two-dimensional case only, as theintegral contains the factor 1 /s . In four dimensions, this factor is 1 /s sothe integral converges for s → ∞ • in the case ℓ = 0 and m = 0, the integral cannot be computed analyti-cally. However, a simple numerical computation shows that the propagatoris finite in the coincident limit for any non-zero ℓ , and this confirms the ul-traviolet regulating nature of the minimal length induced in the field theoryby non-commutativity.Let us now look at the energy density. The Hamiltonian operator for the scalar fieldbecomesˆ H = 12 Z d x h ˙ φ + ( ~ ∇ φ ) + m φ i = 12 Z d~p e − ℓ ( p + ω ) ω (cid:0) ˆ a p ˆ a † p + ˆ a † p ˆ a p (cid:1) . (18)This expression clearly renders unnecessary the normal ordering, usually employedto eliminate the divergent zero-point energy. In fact h | ˆ H | i = e − ℓ m Z ∞ dp e − ℓ p p p + m , (19)which can be shown numerically to be convergent for any m >
0. In the masslesscase, the above integral gives simply (8 ℓ ) − .
3. Particle production
We now turn to the problem of the quantization of the field on a curved space.It is well known that, in a general space-time, one can find more than one com-plete orthonormal sets of modes, and construct the fields as different mode sums.The transformation between two inequivalent basis generates non trivial Bogolubovcoefficients, which can be interpreted as a sign of particle production 38. Let usassume that the Klein-Gordon product (8) can be locally carried over curved space,and let u i and v i be two basis of normal modes of the type (7). The scalar field canbe represented as φ ( x ) = X i ( ˆ a i u i + h . c . ) = X j ( ˆ b j v j + h . c . ) . (20)Here, ˆ a and ˆ b are the inequivalent annihilation operators, which are assumed tosatisfy the usual commutation rules. We can write the damped modes as u i = g u U i and v i = g v V i , where U i = u i ( ℓ = 0) and V i = v i ( ℓ = 0) are the standard modesovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Massimiliano Rinaldi of the commutative theory, and g u,v are the Gaussian damping factors. From whatwe have seen above, the Klein-Gordon product is modified according to ( u i , u j ) = g u ( U i , U j ), together with the analogous expression for ( v i , v j ). Since the modes U i and V j are related by the linear transformations 38 V j = X i ( α ij U i + β ij U ∗ i ) , (21)it follows that v j = g v g u X i ( α ij u i + β ij u ∗ i ) , u i = g u g v X j ( α ∗ ij v j − β ji v ∗ j ) . (22)It is not difficult to prove thatˆ b l = X i ( α ∗ il ˆ a i − β ∗ il ˆ a † i ) , (23)which means that the relation between the two sets of annihilation and creationoperators are not affected by the Gaussian damping factors. Most importantly,we find that, despite the Klein-Gordon product is modified according to (8), theBogolubov coefficients are unchanged with respect to the commutative case, since b jl = − g u g v ( v j , u ∗ l ) ≡ ( V j , U ∗ l ) , (24)and a similar expression holds for α ij . From this almost trivial argument, we deducethat the non-commutative structure of spacetime, when treated in terms of coher-ent states, does not affect the production of particles, given that the Bogolubovcoefficients, and in particular β , are not modified with respect to the commutativecase. However, the energy density associated to these particle is very different whencompared the commutative case. In fact, even though the expectation value of thenumber density operator h N i i = X j | β ij | is unchanged, the energy density associ-ated to it is given by Eq. (18), which, for the i -th particle species can be writtenas ˆ H i = 12 Z d~p e − ℓ ( p + ω ) ω (cid:18)
12 + ˆ N i (cid:19) . (25)Therefore, even if there is a conspicuous production of high frequency particles,their contribution to the total energy density is suppressed.The fact that particle production is not affected by non-commutativity has sev-eral positive consequences. In inflationary models, this result suggests that the spec-trum of primordial perturbations is unchanged despite the high energy modificationson the Feynman propagator. Moreover, gravitons and particles produced during atransition between two different cosmological eras are indistinguishable from theones obtained in the commutative case. Thus, a priori, there are no observationsthat can rule out, so far, this theory.ovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Particle production and transplanckian problem on the non-commutative plane
4. Transplanckian problem
In the case of black holes, it is well-known that the Hawking flux measured by anasymptotic observer derives from an accumulation of in-modes of arbitrarily highfrequency 1. This aspect has always created some unease as it entails arbitrarilyhigh energies, which render dubious the validity of the theory itself. However, non-commutative effects prevent this problem, as we quickly show below.When a free-falling observer near the horizon measures the frequency Ω of thefield modes φ ∼ (4 π Ω) − / exp( i Ω t − i Ω x ) findsΩ ∝ ω (cid:18) − Mr (cid:19) − , (26)where ω is the frequency measured by an asymptotic observer, M is the mass ofthe hole, and r is the radial coordinate of the Schwarzschild metric, which tendto 2 M as we approach the horizon at r h = 2 M
1. The free-falling observer canlocally treat φ as a Minkowski field, thus its energy density is the sum of the energy ~ Ω of each mode. As this diverges when the observer approaches the horizon, oneexpect that this huge energy backreacts against the metric, thus invalidating theinitial hypothesis that the field is a test field propagating on a fixed background.However, as we have shown above, when non-commutativity is switched on, eachmode is equipped with a damping factor of the form exp( − ℓ Ω ) d . Therefore, asthe local frequency increases, the mode and its energy is increasingly suppressed,and the energy density of the field is naturally bounded by the scale ℓ .These results can explain the apparent contradiction between what was foundabove and the results on the Unruh effect 35. In this work, the authors find that thespectrum of the particles seen by a uniformly accelerated detector is suppressed andno longer thermal, when the propagator is modified according to Eq. (14). Thus,in contrast with most models with a minimal length, where the Unruh effect isrobust 10 , ,
39, the Gaussian damping in the propagator has a dramatic impacton the spectrum. This fact can lead to think that if the Unruh effect is calculatedvia Bogolubov transformations, rather than with an accelerated detector, the resultwould be the same as for the commutative case, and hence in contradiction with thefindings on the Unruh effect. However, in this work, what is found to be suppressedis, in fact, the response rate, which is nothing but an energy flux. Therefore thereis no contradiction because here we proved that the energy density of the producedparticles is suppressed, and so it must be also its flux through the detector.
5. Conclusions
In this paper we constructed the field theory of a massive scalar field on the non-commutative plane, by mode analysis. The results coincide with the path integral d For simplicity, we consider massless fields, for which ω = p , as the dispersion relation is notmodified. ovember 9, 2018 0:27 WSPC/INSTRUCTION FILE mpla Massimiliano Rinaldi approach, and clarified the relation between the Euclidean and Lorentzian propaga-tor. We then used this field theory to tackle the transplanckian problem for a blackhole. The main result is that the fuzziness on the manifold puts an upper limit onthe energy density that can be stored near the horizon by a free-falling observer.The quantum backreaction on the geometry of the black hole can in principle becalculated with a suitable effective action, and it represents our next goal. In conclu-sion, we believe that the coherent state formulation of non-commutativity can offernew and intriguing perspectives on the phenomenology of quantum gravity. In thispaper we presented just a glimpse of the potential of this theory, which certainlydeserves further investigations.
Acknowledgments
I wish to thank P. Nicolini, and E. Spallucci for reading the manuscript, and allthe people in the Cosmology Group of the University of Geneva for continuous andstimulating discussions. This work is supported by the Fond National Suisse.
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