Energy Gaps in a Spacetime Crystal

Transcription

1 Energy Gaps in a Spaetime Crystal L.P. Horwitz a,b, and E.Z. Engelberg a Shool of Physis, Tel Aviv University, Ramat Aviv 69978, Israel b Department of Physis, Ariel University Center of Samaria, Ariel 47, Israel Department of Physis, Bar Ilan University, Ramat Gan 529, Israel arxiv: v2 [quant-ph] 8 Nov 29 Abstrat This paper presents an analysis of the band struture of a spaetime potential lattie reated by a standing eletromagneti wave. We show that there are energy band gaps. We estimate the effet, and propose a measurement that ould onfirm the existene of suh phenomena. PACS: 3.3.+p (speial relativity), 71.2.Mq, 71.2.Nr (band struture), w (theory and models), b (rystalline solids), e (miroeletronis) 1 Introdution Based upon work done by Fok[1], Stuekelberg[2][3], and Feynman[4] in the first half of the previous entury, Horwitz and Piron[5], and Fanhi and Collins[6], onstruted a quantum relativisti theory in whih Einstein s ovariant time is onsidered as a dynamial variable. The evolution of a system is then parametrized by a universal invariant τ identified with Newton s time. In this theory, the ovariant wave funtion ψ τ ( x, t), whih evolves aording to the Stuekelberg-Shrodinger equation, is oherent in time as well as spae variables. It provides a simple and straightforward desription of interferene in time[7], in agreement with the reent experiment of Lindner et al[8]. The objetive of this paper is to present an analysis of the band struture of a spaetime potential lattie, reated by a standing eletromagneti wave, using the Hamiltonian originally proposed by Stuekelberg (this Hamiltonian has been shown to lead to the ovariant Lorentz fore[9]), and first-order perturbation theory. 2 Mass Shift in Spaetime Lattie We begin from the eletromagneti, proper-time-independent, Stuekelberg Hamiltonian H = (p µ ea µ ) (p µ ea µ ) 2M whih an be written as (1) 1

2 H = p µp µ 2M ie h M Aµ x µ + e2 2M A µa µ (2) The first term is the unperturbed Hamiltonian (with spetrum orresponding to the mass of the partile). The seond is a perturbation of the first order in A µ, and the third term is a perturbation of the seond order in A µ. We start by finding the first order perturbation of the first order term in A µ. Following methods used in nonrelativisti solid-state physis (see, for example, [1]), our unperturbed wave funtions are ψ k = 1 v e ikσxσ (3) The mass shifts due to the perturbation are found by alulating the eigenvalues of the perturbation operator V k1,k 1 ɛ V k1,k 2... V ɛi = V k2,k 1 V k2,k 2 ɛ... = (4)..... with V k,k = ψkv ψ k d 4 x (5) For the first order term in A µ, this gives us V k,k = ψk ie h M Aµ x µ ψ k d4 x (6) We assume that the eletromagneti wave has the form A µ = A sin (ω γ t) os (k γ z) (7) Then, with V k,k = iek xa h e ikσxσ (8) 4vM ( e i(ωγt+kγz) + e i(ωγt kγz) e i( ωγt+kγz) e i( ωγt kγz)) d 4 x This an be nonzero only when K σ = k σ k σ (9) 2

3 K σ = ω γ k γ ω γ k γ ω γ k γ ω γ k γ (1) We define the edge of a Brillouin zone as the olletion of groups of degenerate states (idential m 2 4 = E 2 2 p 2 z), in whih the distane between the points, in the (E, p z ) spae, an be written as hk γ ( n E Ê + n p ĉp z ), with integer n E and n p. The rank of the edge of the Brillouin zone is determined by filling in the lowest possible absolute values of integer n E and n p. Thus, the edges of the first five Brillouin zones are given by the following (n E, n p ): (n E = ±1, n p = ) or (n E =, n p = ±1) (n E = ±1, n p = ±1) (n E = ±2, n p = ) or (n E =, n p = ±2) (n E = ±2, n p = ±1) or (n E = ±1, n p = ±2) (n E = ±2, n p = ±2) 1st Brillouin zone 2nd Brillouin zone 3rd Brillouin zone 4th Brillouin zone 5th Brillouin zone (11) We see that we an have a mass gap only along the edges of the 2nd Brillouin zone. However, the 2nd Brillouin zone lies entirely on the light one, where it is not likely to find a massive harged partile. 3 Mass Shift at the Edges of the 3rd and 5th Brillouin Zones We now repeat the proess for the first order perturbation of the seond order term in A µ. Our unperturbed wave funtions are as before, so V k,k = 1 ψ e 2 k v 2M A µa µ ψ k d 4 x (12) Substituting eq. 7 for A µ, this beomes V k,k = e2 A 2 e ikσrσ (13) 8vM ( 1 1 ( e i2ω γt + e i2ωγt e i2kγz e i2kγz) 2 1 ( e i(2ωγt+2kγz) + e i(2ωγt 2kγz) + e i( 2ωγt+2kγz) + e i( 2ωγt 2kγz))) d 4 x 4 This an be nonzero only when: 3

4 or or K σ = K σ = 2ω γ 2ω γ 2k γ K σ = 2ω γ 2ω γ 2k γ (14) 2k γ 2ω γ 2k γ 2k γ 2ω γ 2k γ (15) (16) This gives us mass gaps along the edges of the 3rd and 5th Brillouin zone. The 5th Brillouin zone lies entirely on the light one, where we are also not likely to find a massive harged partile, so our main interest lies in the gap at the edge of the 3rd zone, illustrated in figure 1. The edges of this zone are defined by the line equations E = ± hk γ (17) p z = ± hk γ (18) Along these lines, exept for four points (where the edge of the Brillouin zone rosses the light one), the degenerate states ome in pairs, between whih K σ is parallel to either the energy or the momentum axis. Therefore, to find the magnitude of the gap on the edge of the 3rd zone, we solve for the eigenvalues or with V ɛ 3 I = V ɛ 3 I = k 3 k 3 k 3 4β ɛ 3 2β k 3 2β 4β ɛ 3 k 3 k 3 k 3 4β ɛ 3 2β k 3 2β 4β ɛ 3 = (19) = (2) We obtain β = e2 A 2 32M (21) 4

5 Figure 1: Edge of 3rd Brillouin Zone in Spaetime Lattie (within the light one) 5

6 Figure 2: Mass Curve Splitting and Energy Gaps at Edge of 3rd Brillouin Zone ɛ 3± = 4β ± 2β (22) 4 Energy Gaps We assume that the mass of an eletron in the spaetime lattie is onstrained 1 to a narrow range around the mass of a free eletron, M e. The alulated mass gaps will be manifested as a splitting of the mass urve near the edges of the Brillouin zones, resulting in energy and/or momentum gaps, as illustrated in figure 2. The mass equation at the edge of the 3rd Brillouin zone, after the splitting, is ( Me 2 + ɛ 3± ) 2 = E 2 2 p 2 z (23) Substituting the positive p z value in eq. 18, we get (for positive energies) 1 We assume that interation with other fields, as well as self interation, would provide a mehanism for the eletron mass to have stability in the neighborhood of its measured value, in the absene of relatively strong perturbations. A ovariant Lee-Friedrihs model was worked out as a model for a stabilizing mehanism [11]. 6

7 E = (M e 2 + ɛ 3± ) 2 + ( hk γ ) 2 (24) If (as in the estimate below) the seond term in the square root is small ompared to the first one, and ɛ 3± small ompared to M e 2, this beomes E = M e 2 + ɛ 3± + ( hk γ) 2 2M e (25) The vetor potential amplitude A is related to the intensity of the eletromagneti beam reating the lattie as 2I A = (26) ɛ ω γ Combining eqs. 21, 22, 25, and 26, we have E = M e 2 + (2 ± 1) e 2 Iλ 2 16M e 3 + h2 ɛ 2M e λ 2 (27) with λ the wavelength of the beam reating the lattie. This means that all kineti energy values of the eletron between and E = h2 2M e λ 2 + e2 Iλ 2 16M e 3 (28) ɛ E + = h2 2M e λ e2 Iλ 2 16M e 3 (29) ɛ are forbidden inside the lattie. For example, if the eletromagneti beam reating the lattie has a wavelength of 589nm, and an intensity of W m 2, we have Therefore h 2 2M e λ 2 = ev (3) e 2 Iλ 2 16M e 3 ɛ =.5eV E =.5eV (31) E + = 1.5eV meaning that all kineti energies between.5ev and 1.5eV are forbidden. 7

8 5 Disussion and Conlusions We have shown that one an oneive of a spaetime lattie in analogy to the rystals of nonrelativisti solid state physis, assoiated with a standing wave in a avity. The resulting wave funtions, aording to the Shrödinger- Stuekelberg equation, are Bloh waves with energy gaps. Our estimates indiate that this phenomenon an be deteted in laboratory measurements. Referenes [1] Fok, V.A. (1937): Phys. Z. Sowjetunion 12, 44. [2] Stuekelberg, E.C.G. (1941): Helv. Phys. Ata 14, 322, 588. [3] Stuekelberg, E.C.G. (1942): Helv. Phys. Ata 14, 23. [4] Feynman, R.P. (195): Phys. Rev. 8, 44. [5] Horwitz, L.P. and C. Piron (1973): Helv. Phys. Ata 46, 316. [6] Fanhi, J.R. and R.E. Collins (1978): Found. Phys. 8, 851. [7] Horwitz, L.P. and Y. Rabin (1976): Lett. Nuovo Cimento 17, 51. [8] Lindner, F., et al. (25): Phys. Rev. Lett. 95, 441; See also L. Horwitz, Phys. Lett. A 355 (26) 1. [9] Horwitz, L.P. (1984): Found. Phys. 14, 127 [1] Raimes, S. (1961): The Wave Mehanis of Eletrons in Metals (North Holland Publishing Company - Amsterdam), pp [11] Horwitz, L.P. (1995): Found. Phys. 25,