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Konstantin 2 R V ) V  ext | ,where Ψ, Ψ ( | 2 x d t , V is ) 2,3 = r elprmtr.Tewv qaintu becomes thus equation wave The parameters. real are epc to respect hr const( where ndniy h ouino hsdffrnileuto is equation differential this function: of logarithmic solution a The density. on eitoue yaaoywt ovninlsuperfluids. conventional superfluid with In analogy by introduced be non- de- requires full latter a the provide treatment. therefore cannot perturbative vacuum, con- series of not perturbative scription alerts do a a This we has that energy therefore, momentum. us their 8, on were that mention dependence relations Ref. ap- just non-polynomial we of the each here; 4 them approach, in sider different former Sec. in be the derived For to out proach. turn dis- induced Correspondingly, relations an above. persion is discussed spacetime as where phenomenon, theory su- vacuum post-relativistic a perfluid of particle-like framework consider the to within is excitations approach recent, more second, a itntsae sal ubdteLna “roton” Landau the dubbed usually shape, distinct a has us ic h ok fRsnadBayik-iuaand Bialynicki-Birula and Rosen of Mycielski. works the since ours, ntefloigequation following the vacuum in in light of clsuista h rudsaeslto o positive for solution state ground analyt- of the values facilitates that tremendously studies It non-exact ical with non- symmetry. gravity the quantum Lorentz of of nonlinearity generalization theory and logarithmic perturbative development with further a model as a introduces This eetwy.Tefis a st ra aumeet sa as theory. effects Lorentz-covariant vacuum a treat of to is perturbation way small first The ways. ferent func- Gaussian a radius-vector. is a it in of case, wave; tion 3D plane symmetric Broglie de rotationally the a by modulated packet wave c 2 = 35 h iprinrltosfrtescn prahcan approach second the for relations dispersion The hoyo uefli aumcnb omltdi dif- in formulated be can vacuum superfluid of theory A i∂ yi o-oet ii,btohriehas otherwise but limit, low-momenta in ty 003 22)[O:10.1142/S0217751X20400321] [DOI: (2020) 2040032 , t = Ψ . 9979 cleapeo h iprinrelation dispersion the of example ical uhmr ope itr arises picture complex more much a t b to eoe eaiitcwt small with relativistic becomes ation 6,7  ~ ecnfruaeaqatmtheory, quantum a formulate can ne g,Dra 00 ot Africa South 4000, Durban ogy, × − a nw,tog o upssohrthan other purposes for though known, was tlwmomenta, low At . o relation ion ρ e h upeso fdissipative of suppression the res nCreincodnts ti Gaussian a is it coordinates, Cartesian In 2 10 safnto hc osntdependent not does which function a is ) 4 ~ m e nrya ucino momentum of function a as energy He, 10 ∇ 2 ms cm ssoni e.5 hsresults this 5, Ref. in shown As . + V − [IJMPA]) ext F 1 ( ρ shsoial aldthe called historically is ρ ( | = ) F x t , ′ ( ) ρ c b s ) − n( ln | ed to tends = b n( ln ρ/ mc ρ ¯ | ,where ), Ψ s 2 / | 2 ~ c / 0 ρ ≈ ¯ ≈ )  const( c Ψ b where , 8,9 . n ¯ and speed The (1) ρ ), ρ 2

form. As the excitation’s momentum increases, this en- ergy grows from zero, until it reaches a local maximum 1.5 (called the maxon peak), which is crucial for suppress- ing dissipative fluctuations.4 Then it dives down to a lo- cal minimum (called the “roton” minimum, for histori- 1.0 cal reasons), and eventually it starts growing again, all the way to a border of a theory’s applicability domain, akin to a solid curve in Fig. 1a. In the regime of small momenta, the dispersion relation is approximately linear 0.5 with respect to momentum, which is a behaviour typical for relativistic particles.

0 0.5 1.0 1.5

(a) Ep/Ea We thus assume that the physical vacuum has a sim- ilar energy spectrum of excitations - with its own val- ues of parameters of course. It also means that one can 2 use the relativistic approach (with perhaps some tweak- ing by adding additional fields) for a large range of en- ergies - until momentum reaches a value corresponding 1 to the maxon peak, which can be rather large, up to a Planck scale. These intermediate models can still pro- vide valuable understanding about various fundamental 0 phenomena, such as the mass generation mechanism and non-zero extent of particles.5,10–12 However, conceptually new physics steps in near and above the maxon threshold; -1 vacuum Cherenkov radiation and luminal boom being ex- 9 0 0.5 1.0 1.5 amples thereof. 2 2 (b) cp/c0

FIG. 1: Profiles of the energy (2) and corresponding velocity squared, versus p/pa, for the following values of η: 1 (dashed An analytical example of a dispersion relation, which curve), 11.43 (dash-dotted), 20.66 (solid), 25.98 (dash-double- has a Landau form and correct proportions for local ex- dotted). trema, is given by the formula:

2 2 p p 1 −k 1 E = 1+ ηf , f(k) 1 e + 2k D+(k) , (2) p 2M p ≡ 2k2 −ℵ k − a s  a     

where D+(k) is a Dawson function, = √π/2 + c0 = lim cp. As in Fig. 1a, it is the solid curve which is ~ ℵ p→0 1/(e erf(1)) 1.32, pa = 2 /a is de Broglie momentum of interest to us. One can see that at small momenta, its ≈ ~ corresponding to the length scale a = / 2m b , and behaviour follows the relativistic pattern, but becomes a M = p2/2E m. Here, p , E and M set the| | mo- a a a ∝ a a ap rather nontrivial as p grows. Initially it decreases with mentum, energy and mass scales of the theory, whereas increasing p, which corresponds to a photon’s slow-down the parameter η controls local extrema of the dispersion and thus it can be interpreted as a photon acquiring the curve: one can check that Ep acquires the Landau form, effective mass µ – when expanded in Taylor series at small cf. a solid curve in Fig. 1a, only if 11.43 <η< 25.98. p, Eq. (2) expectantly yields a deformed-relativistic dis- Furthermore, Fig. 1b displays the excitations’ veloc- persion: ity c dE /dp divided by its low-momentum value p ≡ p p

2 2 2 2 4 4 6 8 1/2 E = c p + µ (p) c , µ(p) A4(p/p ) + A6(p/p ) + (p/p ) , (3) p 0 0 ≡ a a O a    3

2 where c0 = (pa/2Ma) η (1 /2), A4 = (pa/c0) ( no longer be based on irreducible representations of a −ℵ 2 ℵ− 8/3+2/η)/(2 ), A6 = (pa/c0) (8/5 /2)/(2 ), Poincar`egroup. This region extends until object’s mo- and [kn] denotes−ℵ termsp of order kn and−ℵ above. −ℵ mentum approaches a value corresponding to the “roton” NoteO that this effective mass generation mechanism minimum, after which the maximum attainable veloc- is different from the gap mechanism in conventional ity can eventually go above c0. This is where the lumi- quantum condensed matter such as superconductors: it nal boom occurs, which is the vacuum analogue of sonic is a post-relativistic effect and it happens before one boom in air.9 Beyond that value, breakdown of the quan- reaches any of the local extrema. Since c0 c, the fol- tum liquid occurs: a moving object or observer no longer lowing experimental constraint applies to our≈ parame- experiences vacuum condensate nor induced spacetime. −38 ters: aMa/√η = (~/c0) 1 /2 2.04 10 g cm, −ℵ ≈ × whereas A6 does not explicitly depend on η. p A series (3) converges only at p

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