A Note on Massless Quantum Free Scalar Fields. with Negative Energy Density

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1 Adv. Studies Theor. Phys., Vol. 7, 13, no. 1, HIKARI Ltd, A Note on Massless Quantum Free Scalar Fields with Negative Energy Density M. A. Grado-Caffaro and M. Grado-Caffaro Scientific Consultants, C/ Julio Palacios 11, 9-B, 89-Madrid, Sain ma.grado-caffaro@saienzastudies.com Coyright 13 M. A. Grado-Caffaro and M. Grado-Caffaro. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Abstract We obtain an imortant result related to the negative local energy density of a massless quantum free scalar field in the context of quantum inequalities. In fact, considering an inertial observer on a world-line of the Minkowski sace-time and a massless quantum free scalar field such that its energy density can take negative values, an interesting theorem is derived by choosing truncated Dirac s delta function as samling function relative to two key quantum inequalities. By the above theorem, we obtain a lower bound for the average value of an involved relevant function of olynomial growth. PACS: 3.7.+k; 4.6.+v; 98.8.Qc Keywords: Negative energy density; Massless quantum free scalar field; Quantum inequalities; Samling function; Minkowski sace-time

2 55 M. A. Grado-Caffaro and M. Grado-Caffaro 1. Introduction Energy density is classically oint-wise non-negative but can take negative exectation values after quantization although the satially integrated density retains its natural nonnegativity. Consider, for instance, backflow by which the state of a quantum article with one-dimensional motion is a suerosition of right-moving lane waves. It has been shown that the robability of finding the article in the right-hand half-line may decrease, i.e., the robability flux may be negative desite that the exectation value of the momentum, as well as any ower of it, of the article is non-negative. In quantum mechanics and quantum field theory as well as in quantum otics and condensed matter hysics [1-4], interesting examles uon negative energy density can be found although the corresonding satially integrated density, that is, the energy, remains ositive. In articular, quantum field theory is a domain where crucial questions are certainly linked to roblems related to negative local energy density. Quantum field theory resents examles in which the satially-averaged energy density of a given scalar field, since it is continuous, then takes both strictly ositive and negative values and can also vanish. On the other hand, all the ointwise energy conditions of classical general relativity do not hold in quantum field theory. In general relativity, a classical field satisfies the so-called weak energy condition after which the energy density is everywhere non-negative as measured by every observer. But, as in backflow, in quantum field theory we areciate that the negativeness of the energy density is not arbitrary at least for inertial observers. As a matter of fact, certain restrictions to (in rincile) arbitrarily negative energy density are governed by the socalled quantum inequalities. In this resect, the urose of the resent note is to rove a theorem based uon two of these inequalities in relation to a given massless quantum free scalar field measured on an inertial world-line of the Minkowski sace-time. Our formulation oens new avenues to treat satisfactorily the main roblems of the subject in question which, to date, has been tackled in wrong aers of an areciable art of the current literature as, for instance, ref.[5] where manifestly wrong aroaches have been done.. Theory First we regard the following quantum inequality for a quantized free scalar field measured over an inertial world-line of the Minkowski sace-time (see, for instance, ref.[1]):

3 Note on massless quantum free scalar fields 551 ρ (1) () t f ( t) dt ( ω) f ( ω) dω where ρ () t is the time-deendent exectation value of the local energy-density oerator T ˆ ( t,) of the field ( T ˆ ( t,) ρ( t) ; ρ ( t ) refers to every admissible Hadamard state). In other words, ρ ( t) is the satially-averaged time-deendent energy density of the field; ρ () t can be, aart from to be null, either strictly ositive or strictly negative. On the other hand, f ( t) is the so-called samling function which can be every smooth (strictly ositive) real-valued function of time, ( ω) is a strictly ositive real-valued function of olynomial growth, and f ( ω) is the Fourier transform of f () t, that is, f ( ω) f ( t) ex( iωt)dt. From this definition, it is clear that ω is angular frequency. From now on, we will assume ρ ( t) as a negative quantity and we will also remove the usual assumtion done in the current literature by which f () t must have comact suort. On the other hand, it is well-known that any smooth and comactly suorted function is square integrable so the left-hand side of (1) has full sense. Now, extraolating inequality (1) to finite intervals of time and frequency, we have: Lemma 1. Let us consider a quantum free scalar field measured over an inertial worldline of the Minkowski sace-time so that the time range is finite namely t while the angular-frequency range is ω π, being the characteristic width of the samling function () t f which now is assumed to be square integrable on [, ] but not necessarily comactly suorted. This function can be either a function in the strict sense or a generalized function. In addition, let us assume that ρ ( t) verified:. Then it is ρ() () π t f t dt ( ω) f ( ω) dω () Furthermore, one has:

4 55 M. A. Grado-Caffaro and M. Grado-Caffaro Lemma. Under the same hyothesis of Lemma 1, regarding also that the field is massless, then there exists a minimal strictly ositive real constant denoted by c such that: c ρ() t f () t dt (3) d where d is the dimension of the Minkowski sace-time. Given the above two Lemmas, now we may enunciate: Theorem. If the field in question is also massless, the following inequality is satisfied: c π d 1 where denotes the average value of ( ω) on [, π ]. Proof: We choose f () t δ ( t ) t [, ] and f ( t) t >, where δ () stands for truncated Dirac s delta function such that δ ( t) = 1 if t = and δ () t = if t. Inserting this generalized function namely f ( t) into inequality () (Lemma 1), since δ ( ) = 1, then the left-hand side of () becomes ρ ( ) δ ( ) = ρ( ) ; moreover, we have that f ( ω) ex( iω ) so f ( ω) 1. Consequently, we get: (4) ρ ( ) π ( ω) dω (5) The average value of ( ω) on [, π ] reads: π π ( ω) dω (6) From the conjunction of (5) and (6), it follows: ρ ( ) π (7) On the other hand, considering now the field as massless, we relace f () t δ ( t ) into (3) (Lemma ) so we find:

5 Note on massless quantum free scalar fields 553 c ρ( ) (8) d By Lemma, the constant c is minimal which means that the right-hand side of (8) is smaller or equal than the right-hand side of (7) so inequality (4) is satisfied 3. Conclusions In order to rove the receding Theorem, as a singular fact, we have emloyed a generalized function as samling one. On the one hand, it is clear that this function (truncated Dirac s delta function) is square integrable and, on the other hand, its suort is non-comact since the suort of the generalized function in question is the set { } which, although trivially bounded, is not closed. In this resect, notice that, on R n, (here, of course, we have that n = 1 ) for a set to be comact, it is necessary and sufficient that the set be closed and bounded. Finally, we wish to remark the role of the constant c as minimum strictly ositive real value indeendent of. References [1] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical analysis on the energy density of a laser field measured along an inertial world line of the sace-time, Otik 1 (11), [] S. Kruchinin, H. Nagao, S. Aono, Modern asects of suerconductivity: theory of suerconductivity,. (World Scientific Pub. Co. Pte. Ltd., 1). [3] S. Kruchinin, H. Nagao, Nanoscale suerconductivity, Int. J. Mod. Phys. B 6 (1), [4] N. Bogolubov (Jr.), S. Kruchinin, Modern aroach to the calculation of the correlation function for suerconductivity model, Mod. Phys. Lett. B 17 (3),

6 554 M. A. Grado-Caffaro and M. Grado-Caffaro [5] S.P. Eveson, C.J. Fewster, R. Verch, Quantum inequalities in quantum mechanics, Ann. Henri Poincaré 6 (5), 1-3. Received: Aril 5, 13