Quantum mechanics with geometric constraints of Friedmann type

1 Introduction

In the first formulation of Mach’s principle, it is stated that inertial frames of reference in the whole space are connected with the distribution of matter in the Universe. It should be noted that to the first approximation the inertial coordinate system is spatially connected not with the absolute space, but with the motionless sky of stars. Along with that, the absolute space is the abstraction which cannot be revealed empirically. Moreover, according to Mach, Newton’s first law can have a limited value only. The latter statement is considered to be false. In work [1], it is demonstrated that in geometrodynamics at a quantum level, the law of inertia of a Planck particle is substantially modified in the quantum epoch of the Universe evolution. The Planck particle left to itself, acting upon itself gravitationally, moves in superspace-time non-uniformly and without being in a quiescent state. It means that the Mach’s principle concerning the limited value of Newton’s first law is correct.

In quantum geometrodynamics [2–15] based on the general theory of relativity, time vanishes. Inevitability of time vanishing can be often explained by the fact that time cannot be correlated with the operator. In this respect in works [16–18], it is demonstrated that in quantum theory if − ∞ < E < +∞ and while E₁ ≤ E ≤ E₂ for the systems with a continuum energy spectrum, a self-adjoint time operator t^=iℏ∂∂E can be introduced. Only if 0 ≤ E < +∞, it is not self-adjoint, which is not a problem as the mathematic properties of such a time operator and with 0 ≤ E < +∞ are sufficient to consider time a quantum-mechanical observable (for the systems witha continuum energy spectrum, time is also considered as a quantum-mechanical observable). It means that time classicality is not treated any more as an argument for the existence of a usual “observer” in quantum geometrodynamics. On this basis in work [5], the possibility of time creation is found where time is something that Lemaitre-Friedmann atomic clock shows. Therefore, the possibility of time creation is the possibility of such clock “assembling” in a quantum process of Planck particle tunneling through a potential barrier in superspace. This quantum process runs for zero time so that the time is not required for the description of time creation. In the first experiment, time creationtakes place while other experiments of the series run in time as the possibility of creation implies a series of test trials which should run in time.

The identical Hamiltonian equality to zero gives rise to the problem of time in quantum geometrodynamics based on general relativity theory (GRT). In the latter, time is either introduced quasi-classically (the Dirac–Wheeler–De Witt approach) or using a gauge condition (the Arnowitt–Deser–Misner approach based on the reduction to physical variables for which the Hamiltonian is nonzero) setting the reference system, which leads to the following problems: physical predictions may depend on the choice of the gauge condition; there arises a gravitational analog of the problem of Gribov copies, the latter does not allow us to use time-setting gaging in the entire superspace. The quasi-classical way of introducing time, however, physically implies that the gravitation background plays the function of time variable, the quantum properties of the background being neglected, and the material fields alone being quantized. Within the Dirac–Wheeler–De Witt formalism, the absence of positive certainty of the scalar product makes the possibility for probabilistic interpretation of the wave function difficult.

In quantum geometrodynamics based on relativistic theory of gravitation with Logunov constraints Dμg~μν=0 asymptotically combined with Gliner’s idea of the material interpretation of the cosmological constant in a classical potential form [5], time does not vanish. Within this approach, the field equations of the theory are derived from the constrained extremum of the action with constraints Dμg~μν=0, because of which scalar-gravitational Hamiltonian is nonzero. Consequently, it solves the problem of time in quantum geometrodynamics, since instead of the Wheeler–De Witt equation

there occurs the following equation possessing reparametrizing invariance with respect to the class of replacement of the time coordinate x₀:

the corollary of which is the dependence of Ψ on the physical time t∂g00∂x0=∂∂t and, in the stationary case, there occurs a gravitational analog of the conventional equation ~^HΨ = 0 describing the Lemaître primordial atom instead of the equation ~^HΨ=EΨ,~^H=VU0−ε^a,V=4πa33,Vε^a=π^a22Mp2a,π^a=−i∂∂a. This is due to the fact that two-dimensional superspace-time {a,t} is a flat with a metric dσ² = c²dt² − da², where dt=aa03dx0,a– a scale factor, which contains the information about the curvature of the effective Riemannian space {x₀, x₁, x₂,x₃} with a metric dS2=aa06dx02−aa02dl2. Therefore, in quantum theory developed in such superspace-time the Hamiltonian should not be equal to zero since the corresponding equation of quantum geometrodynamics is not required to possess symmetry of General Relativity

For a uniform isotropic constant scalar field with the potential energy density U (ϕ) = U₀ < 0 the gravitational analog of the stationary Schrödinger equation becomes

where H^a=−π^a22Mp2a−VU0. We managed to solve the equation in the analytical form as

Here x=23H0Mp2a3,H0=8πGU03,ν=−13,Lnνx is the generalized Laguerre polynomial, N=(9α)1/6Γ(13)Γ(23)Γ(23+n) is the normalization, and α=2H0Mp2. The quantizing condition is written a

To reveal special features of the physical content of the model, the above gravitational analog of the Schrödinger equation can be written as a Dirac condition

Here H^D=∂22Mp∂a2+W(a),W(a)=MpaE−VU0. In the Lemaître primordial atom, a Planckian particle plays the role of an electron, and the effective potential is as follows:

The effective potential of the gravitational analog of the stationary Schrödinger equation written in the form of a Dirac one exhibits an extraordinary property due its parametric dependence on the energy E. ForE > 0, U₀ > 0 the potential is the barrier W (a, E) = MpaE−4πU03a3, whereas for E < 0, U₀ < 0 – the potential well W (a, E) = Mpa−E+4πU03a3, with increase in |E| the width and depth of the potential well (the width and height of the potential barrier) also increase. From the mathematical point of view, the special features of the potential and the gravitational analog of nonstationary perturbation theory result from the form of the kinetic-energy operator T^a=12Mp2a∂2∂a2. The wave functions of the Lemaître atom are normalizable and form a complete function system, however, similar to the basis of coherent states they are not orthogonal. Therefore, the Lemaître primordial atom, in contrast to an ordinary atom, can radiate permanently. Physically, the permanence is due to the fact that the width and depth of the potential well W (a,E) = Mpa−E+4πU03a3 increase with increase in |E|– “the Planckian particle itself is digging out a potential well for itself”.

On this basis the Wheeler program extended to time and allowing determination of mass without mass, charge without charge, spin without spin, time without time has been realized on the geometrodynamical basis, taking no account of the features of topological character. With the help of the Dirac procedure for extracting the root of the Hamiltonian operator of the free gravitational field a quantum geometrodynamics with fractional spin in flat superspace-time can be constructed. The charge and time are born in the process of the Planckian particle tunneling through the potential barrier in superspace. The problem formulated by Einstein concerning determination of the inertial mass using the curvature of the effective Riemann spacetime due to the constant scalar field has been solved. For the flat Universe the Mach principle has been revived as follows: if there is no constant scalar field (there is no curvature of the effective Riemann space) – there is no inertial mass [5].

In this respect we consider the influence of gravitational constraints on the law of inertia at a classical and quantum levels and on the problem of time in the usual quantum mechanics.

2 Classical and quantum inertia with constraints

The influence of the constraints Friedmann type φx,x′=x′2a02−x2b02−k=0,k = 0, ± 1, a₀, b₀ – constants on the dynamics of a free particle is investigated. In a phase space {x, x′}, these constraints φx,x′=x′2a02−x2b02−k=0, while k = ± 1 are canonical equations of hyperbola or equations of the second-orderhyperbolic cylinder in a phase space {x, x′, z}.

For the investigation the variational calculus is applied. From the action extremum condition with constraints δS = 0 Lagrange equation follows

where Lagrange function is L (x, x’) = mx′22+λφx,x′. Let us consider the dynamics of a free particle with hyperbolic constraint φ (x, x′) = x′2a02−x2b02−k=0. In such a case, Lagrange equation takes the form

The solution of the system of equations (1) with hyperbolic constraints is infinite and takes the form

Where a0=x~0H0,b0=x~0,λ=−ma024,x~0,H0− integration constants. It should be noted that the system has a trivial solution describing the quiescent state. We can assume that for the constraint φ_i (x, x′) = x′2a02−1=0, the system of equations (1) has a Newton’s inertial solution of x = v₀t.

It should be also noted that if the variable x is equal to a scale factor a and the constant of integration which has a frequency dimension H₀ (ħ = c = 1) is equal to the Hubble value H0=8πGU03, then equations (1) are similar to Friedmann equations for a de Sitter universe filled with vacuum-like environment with the equation of state ε + p = 0, p = − U₀ and with the metric dS² = N² (dx⁰)² − a2dr21−kr2+r2dϑ2+sin2ϑdφ2 and with time dt = Ndx⁰, where U₀ > 0– density of the potential energy of the scalar field:

While v₀ = c, these equations are equal to Friedmann equations, where c is the speed of light. In the general theory of relativity, the case k = 0 corresponds to the flat Universe, k = ± 1 corresponds to the closed and open Universe. Inertial constraint φix,x′=x′2a02−1=0 in the theory of relativity corresponds to the quintessence with the equation of state p=−ε3.

Let us consider the dynamics of a free particle with elliptic constraint φ (x, x′) = φx,x′=x′2a02+x2b02−1=0, which is similar to Friedmann energy equation with the density of the potential energy ε = U₀ < 0. In such a case, Lagrange equation takes the form

The solution of the system (2, 3) is finite x = x~0 sin (H₀t), λ=−ma024,a0=x~0H0,b0=x~0. From the equations (2, 3), it follows ddtmx′22+mH02x22=0 so that mx′22+mH02x22=mH02x~022=E=const

For the elliptical constraint, the Hamiltonian corresponds to the constraints and is equal to

where h=π22m+mH022x2,π=∂L∂x′=2m+2λa02x′=mx′− generalized momentum of the particle.

It should be noted that for the hyperbolic constraints the value of the path curvature inverse of the radius of the circle of curvature in the phase space is nonzero

where x′=dxdt,y′=dydt,y=dxdτ,τ=H0t. For the elliptic constraint k0=−1x~0. For the inertial constraint φ_i (x, x′) = x′2a02−1=0, a path curvature k₀ = 0. This means that the potential term in the Hamiltonian (4) is determined by the path curvature in the phase space. Obviously, a curved line in a flat phase space can be interpreted as a geodesic line in the corresponding curved phase space. At the same time it means that in such a case the constraints can modify Newton’s first law.

Since equality (4) H = 0 is equivalent to equality h = E, then from the expression (4), the unsteady-state equation follows

where h^=π^22m+mH022x2,Ψx,t=e−iEtΨx, so that owing to the energy constant mH02x022=E=const time does not vanish in quantum mechanics with constraints φx,x′=x′2a02−x2b02−k=0, while k = ± 1 and φx,x′=x′2a02−x2b02−1=0. Time vanishes only for the constraint φx,x′=x′2a02−x2b02−k=0 if k = 0, since in this case instead of the equation (5) occurs equation ĥ Ψ (x,t) = 0, h^=π^22m−mH022x2.

The solution of equation (5) and the condition for quantization have a known form

where N₀ – normalization constant, H_n (ξ) – Hermitian polynomial, ξ=xα0,α0=ℏmH0.

For the hyperbolic constraints φx,x′=x′2a02−x2b02−k= 0 if k = ± 1 Schrodinger steady-state equation has a form

so that in this case the potential term of geometric origin determined by the hyperbolic constraint −mH022x2 is the barrier while the energy spectrum is continuous.

In this case, quantum phenomena can occur as well. For example, a barrier equation (6) takes place in the theory of particle production from vacuum by intensive homogeneous electric field. Functions (7) essentially coincide with the solutions leading to Klein paradox. It implies that the current of the passing particles is higher than the current of the falling particles by the potential barrier with the height > 2m₀c². The exceedance of current, mentioned by Klein, is determined by the increase of the whole number of the particles as a result of pair production by the field of the barrier.

This process is investigated within the field theory [19]. In case of the potential barrier, equation (6) in dimensionless variables takes the form

The solutions of equation (7) are the function of parabolic cylinder

For the equation

the solutions take the form

For the function of the parabolic cylinder there is a relation

The scalar product of such solutions is the Wronskian determinant

so, that if ν=−12+iμ2,z=−1−iξ

The asymptotics of a function D−12+iμ2−1−iξforμ>>1 have the form

where φξ=π8i+μ+iμ4ln⁡2+iμ2lnξ.

This means that the function D−12+iμ2−1−iξ corresponds to a positive spatial frequency solution for ξ → ± ∞, i.e. antiparticle. The complex conjugate solution corresponds to a negative spatial frequency, i.e. particle. This means that, although the field exists everywhere, it is possible to specify regions of space in which the states of particles and antiparticles can be determined. From the asymptotics it can be obtained that with μ ≥ 1 the length of the formation of the process Δx ≈ μα₀, and with μ < 1 this length equal to Δx ≈ α₀ [19].

From the asymptotics of the functions of a parabolic cylinder it follows that ^(±)Ψ (ξ) = NνD−12∓iμ2−1±iξ±Ψξ=NνD−12±iμ21∓iξ are the solutions of equations (5, 7) thathavepositiveornegativefrequencies corresponding to the index (+) and (−), while ξ → − ∞ ( ξ → + ∞). The functions _(±)Ψ (ξ) satisfy normalization and orthogonality conditions

The similar conditions are also for the functions ^(±)Ψ(ξ).

There are the following relations for these solutions

From (10, 11) it follows

where

From normalization conditions ±Ψ,±Ψ≡±Ψ∙ξi∂↔∂x±Ψξ=±1 we can obtain +Ψ,+Ψ=c1∙=c′2Δ,+Ψ,−Ψ=−c2∙=−c′1Δ,c12−c22=1, so that

The relations similar to (10 - 12) can be obtained for the creation operator a^+,α^+β^+ and annihilation operator a^,α^β^ of particles (antiparticles). It is true that from (10 - 12) and from the relation Ψ=a^+Ψ+b+−Ψ=α^+Ψ+β++Ψ there can be obtained the relations

From the normalization conditions with account of (9), it follows that the normalization factor is N_ν = α0e−πiν+12424=e−πμ8α024. It is easy to demonstrate that the orthogonality condition is met ∓Ψ∙ξi∂↔∂x±Ψξ=0. In case of ν=−12+iμ2,z=−1−iξ the lowering (a) and raising (b⁺) operators can be expressed in a differential form

Then aa+−a+a=bb+−b+b=i,ab++b+a=2ν+1=iμ so that the equation d2dξ2+μ+ξ2Ψνξ=0 can be expressed as H^Ψνξ=2ν+1Ψνξ, where the operator is H^=ab++b+a.

The average number of scalar particle pairs generated from the vacuum state 0=N0e−z24 is determined by the relation 0α^+α^0,wherea^0=0,0a^+=0. If according to the relations (13), α^=c1a^+c2∙b^+,thenα^0=c2∙b^+0,0α^+=c20b^. In this case the average number of scalar particle pairs is

where 0bb^+0=i. From relation (8, 10), it follows that c1=2πe−π4μ+iΓ12+iμ2,c2=e−π2μ+i, so that the average number of scalar particle pairs produced by the barrier is

Therefore, the average number of pairs generated per unit time P=N¯T=H04π2e−2πm0H0 where the function of a homogeneous electric field is performed by the value determined by the Hubble constant E~=m0H02e=m02e8πGU03. As this takes place, in the corresponding notations the probability takes the standard form [19] P=H04π2e−2πm0H0=χ2π2τ0e−πχ,χ=E~E~0,E~0=m02c3eℏ,τ0=ℏm0c2.

3 Conclusion

In case of the constraint φix,x′=x′2a02−1=0, the particle left to itself in the coordinate space is motion less or moves uniformly. In case of the constraint φx,x′=x′2a02±x2b02−k= 0, a free particle in the coordinate space is motionless or moves non-uniformly owing to the effective force determined by the constraints. It implies that Mach’s principle concerning the limited value of Newton’s first law is correct at a classical and quantum levels. Time does not vanish in quantum geometrodynamics with Logunov constraints as well as in quantum mechanics with constraints φx,x′=x′2a02±x2b02−k=0. Time vanishes in ordinary quantum mechanics with constraints φix,x′=x′2a02−1=0.