Second Quantization of Time and Energy in Relativistic Quantum

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2 First quantization: conﬁguration, momentum, time and energy representations[8]

Canonical quantization and factorization of the special relativity free particle µ 2 2 invariant p pµ = π := (m0c) yields a constraint that is satisﬁed by the linear equation: [γν pˆ m c] Ψ = 0 (1) ν − 0 | i provided that in the Minkowski metric ηµν = diag(1, 1, 1, 1) the following anticommutation and commutation relations are satisﬁed:− − −

γµ,γν =2ηµν I [ˆp , pˆ ] = 0 (2) { } µ ν where I is the 4 4 identity matrix. Thus the γ’s satisfy a Cliﬀord algebra and are represented× by matrices. Eq.1 is recognized as the Lorentz invariant Dirac the introduction of an operator t is basically forbidden and the time t must necessarily be considered an ordinary number (‘c’ number) in Quantum Mechanics” 3A. Zee, ”Quantum Field Theory in a Nutshell”, Princeton University Press (2010)

2 equation with:

2 i 0 i 0 HD = cα.ˆp + βm0c , α = γ γ β = γ

In the same way, canonical quantization and factorization of the special µ 2 2 relativity invariant x xµ = s := (τ 0c) yields a constraint that is satisﬁed by the linear equation: [γν xˆ τ c] Ψ = 0 (3) ν − 0 | i provided now that:

γµ,γν =2ηµν I [ˆxµ, xˆν ] = 0 (4) { } Eq.3 is a Lorentz invariant equation satisﬁed by the self-adjoint time operator:

i 0 i 0 T = α.ˆr/c + βτ 0 α = γ γ β = γ introduced earlier in analogy to the Dirac Hamiltonian[9]. It introduces an intrinsic time characteristic τ 0 of the system. µ µ Finally the purely imaginary symmetrized invariant O− := (ˆx pˆµ pˆµxˆ ) −ν that satisfies Born’s reciprocity invariance under the transformationx ˆ pˆν , pˆ xˆν [10] suggests using Planck’s constant to accept: → ν →− xˆ0pˆ pˆ xˆ0 =x îpˆ pˆ xî = i~ (5) 0 − 0 i − i that insures the satisfaction of the constaint:

[ˆxµpˆ pˆ xˆµ] Ψ = [(ˆx0pˆ pˆ xˆ0) (ˆxipˆ pˆ xî)] Ψ =0 µ − µ | i 0 − 0 − i − i | i µ ν Eqs.5 complement the commutation relations [ˆpµ, pˆν ]=0and [ˆx , xˆ ]=0to yield: a) infinite continous range of the four-space and four-momentum spectra; b) the known configuration and momentum representations of the operators ν xˆ andp ˆν ; c) the Fourier transfom relation between the representations of the system state vector and d) the position-momentum uncertainty relation[8]. In the configuration representation, Ψ(r,x0) = x Ψ , defining t := x0/c, Eq.1 reads: h | i ∂Ψ(r,t) ∂ i~ = i~cαj + βm c2 Ψ(r,t) (6) ∂t {− ∂xj 0 } recognized as the time dependent Dirac equation for free motion. In the momentum representation Φ(p,p0)= p Ψ , defining e := cp0, Eq.3 reads: h | i ∂Φ(p,e) ∂ i~ = i(~/c)αj + βτ 0 Φ(p,e) (7) ∂e { ∂pj } that clearly relates energy changes to momentum changes. As a self-adjoint operator, T is the generator of infinitesimal momentum displacements (Lorentz boosts) δp = (δe/c)α =(δe/c2)cα and thus indirectly energy displacements, circumventing Pauli’s objection. In the same way, HD is the generator of infinitesimal space displacements δr = cα(δt) where cα =dr/dt

3 is the velocity operator[15]. For a wave packet cα = vgp, the group velocity. Thus δp = (δm)v = γm v and δr = v h(δt)i. h i gp 0 gp h i gp T also generates a phase change δϕ = β(δe)τ 0/~ while HD generates a 2 2 phase change δχ = β(δt)m0c /~. These are equal provided δe = m0c and 2 δt = τ 0. Furthermore a common ﬁnite 2π phase shift requires τ 0 = h/m0c . 2 In conclusion, the dynamical time operator T = α.ˆr/c + βh/m0c (where the 2 parameter τ 0 is equated to de Broglie period h/m0c ), generates the Lorentz boost that gives rise to the de Broglie wave[18]. This also supports the de Broglie period as an intrinsic property of matter, in agreement with experiment[16]. These Dirac energy and time operators satisfy the commutation relation: [T,H ]= i I +2βK ~ +2β τ H m c2T (8) D { } { 0 D − 0 } where K = β 2s.l/~2 +1 [15] is a constant of motion. The related uncertainties { } are such that (∆T )(∆HD) (∆r)(∆p)[9, 13], sustaining the interpretation given by Bohr originally: the≃ time uncertainty in the instant of passage at a certain point is given by the width of the wave packet which is complementary to the momentum uncertainty and thus to the energy uncertainty[14]. In the Heisenberg picture:

dT (t) 1 2 = [T,H ]= I +2βK + β τ H m c2T (9) dt iℏ D { } iℏ { 0 D − 0 } that upon integration yields: t T (t)= I +2βK +2β(1/i~)τ H t (1/i~)m c2β dtT (10) { 0 D} − 0 Z0 as HˆD is constant. The last term introduces an oscillatory behavior (Zitterbe- wegung) about a linear time evolution. The eigenspinors of the self adjoint energy HD and time T operators (Appendix A) provide orthogonal basis additional to the continous vector ones generated by 4 operatorsx ˆν andp ˆν , with the following characteristics. The energy spectrum 2 goes from to + , with a 2m0c gap at the origin. The time spectrum −∞ ∞ 2 goes from to + , with a 2τ 0 gap at the origin. As τ 0 = h/m0c (the deBroglie or−∞ Compton∞ period[17, 18]) the gaps are seen to be complementary. To a small energy gap corresponds a large time gap, and viceversa. This com- plementarity may provide the first indication that the electron neutrino mass has to differ from zero as this would send the time gap to infinity. Also in the 2 same way that the energy spectrum clearly defines non relativistic (cp m0c ) and relativistic energy limits (cp m c2) , the time spectrum recognizes≪ short ≫ 0 (r/c τ 0; r cτ 0) and long (r/c τ 0; r cτ 0) time or space limits. As in≪ the non relativistic≪ limit Eq.6 yields≫ the two-component≫ positive energy Schrödinger-Pauli equation[15], Eq. 7 results in a two-component positive time short range approximation. 4This clarifies the confusion addressed by Hilgevoord[21] between the coordinates of a point in space and the position variables of a particle (”clearly fostered by the notation x,y,z for 0 both concepts”); as well as noting that t := p /c and HD are not canonical conjugate variables.

4 3 Second quantization of the energy-momentum and time-space ﬁelds

(Note There are many books on quantum field theory. In this section the presentation of F. Schawbl, ”Advanced Quantum Mechanics”, Chapter 13[19] is followed, where the representation of the field is given as a superposition of free solutions in a finite volume V . The passage to infinite volume is achieved with 1/2 3 m d k √m 3 where the factor √m is chosen in order to cancel k V ek ⇒ (2π) k0 Pthe factor 1/√m Rin the spinors, so that the limit m 0 exists). a) The time-space representation → The second quantization of the Ψ(r,t) field considers its expansion in terms of the spinor eigenvector basis eq (Appendix A) and transforms the expansion coefficientes into creation and{| i} anhilation (particle and antiparticle) operators:

2 1/2 m0c ep ip.r/~ ep ip.r/~ r Ψˆ = Ψ(ˆ r)= (ˆb pu (p)e− + dˆ† w (p)e ) (11) | Ve q q qp q p,q p D E X where the ”energy spinors” are:

eq = uep (p) p e >m c2 and eq = wep (p) p e

2 1/2 ep + m0c χq ep σ uq (p) = c .p (13) 2m c2 2 χq 0 ep+m0c 2 1/2 cσ.p ep + m0c 2 χ wep (p) = ep+m0c q (14) q − 2m c2 χ 0 q 1 with p = p , e = + (cp)2 + (m c2)2 and for q = 1, 2 χ = , | | p o 1 0 0 p χ = . Thus u (p) correspond to positive energy and up and down 2 1 1,2 spin, while w1,2(p) correspond to negative energy and up and down spin, as ep can be seen clearly in the rest frame where p = 0. The minus sign in wq (p) insures that the charge conjugation operation C transforms the spinors uq(p) into wq(p) and viceversa. It then follows:

µ 3 µ µ P = i d x ψγ¯ ∂ ψ = p (ˆb† ˆb p dˆ ˆpdˆ† ) (15) { 0 } qp q − q qp p,q Z X where: 0 P = cp (ˆb† ˆb p dˆ ˆpdˆ† )= e (ˆb† ˆb p dˆ ˆpdˆ† ) (16) 0 qp q − q qp p qp q − q qp p,q p,q X X

5 Anticommutation of the ﬁeld operators to satisfy FermiDirac statistics[20] and normal ordering to avoid zero point terms yields:

0 ˆ ˆ ˆ ˆ P := H = ep(bq†pbqp + dq†pdqp) > 0 (17) p,q X i.e., the total energy as the sum of positive energy quanta ep for both states, now interpreted as representing electrons and positrons respectively. The total momentum is given as: ˆ ˆ ˆ ˆ P = p(bq†pbqp + dq†pdqp) (18) p,q X As components of a four vector, P 0 and P i are part of the energy-momentum tensor (stress-energy tensor):

T µν = iψγ¯ µ∂ν ψ (19) whose other components yield the total (orbital plus spin) angular momentum of the system. The momentum operator P µ is furthermore shown to be the generator of space-time displacements, i.e.:

µ µ iaµP iaµP e Ψ(x)e− = Ψ(x + a) (20)

b) The energy-momentum representation The above procedure can also be applied to the energy-momentum representation Φ(p)= p Ψ = Φ(p,p0) and its adjoint. Expanding the ﬁeld Φ(p, E) in the spinor ht | basisi (Appendix A) yields5: {| ri}

1/2 τ 0 tr ip.r/~ tr ip.r/~ p Ψˆ = Φ(ˆ p)= (ˆa ru (r)e− +ˆc† w (r)e ) (21) | Vt q q qr q r,q r D E X with eigenvectors, now ”time spinors”:

1/2 χ tr tr + τ 0 q uq (r) = σ.r/c ...t> 0 (22) 2τ 0 χ tr+τ 0 q ! 1/2 σ.r/c t tr + τ 0 χ w r (r) = tr+τ 0 q ...t< 0 (23) q 2τ χ 0 q ! 1 0 with t = + (r/c)2 + τ 2 and again q = 1, 2 , χ = , χ = . r 0 1 0 2 1 tr tr Thus u1,2(r)p correspond to positive time and up and down spin, while w1,2(r) correspond to negative time and up and down spin.

5Here a ﬁnite momentum volume is considered, yielding an unfamliar discretization of space. However in the inﬁnite volume limit Pr goes to the familiar R dr.

6 One now obtains:

µ µ T = (x /c)(ˆa† aˆ r cˆ rcˆ† ) (24) qr q − q qr r,q X As before, requiring anticommutation and normal ordering yields:

0 2 2 T = tr(ˆaq†raˆqr +ˆcq†rcˆqr) > 0 tr =+ (r/c) + τ 0 > 0 (25) r,q X q i.e., the time operator ﬁeld contains only positive times. This development in- 2 2 troduces time quanta tr =+ (r/c) + τ 0 > 0 that are created and destroyed. by the operators a ,a and c ,c . In analogy to Eq.17, Eq.25 represents a total † p† intrinsic time associated with the system. One also obtains from Eq.23 the vector relation:

T = (r/c)(ˆaq†raˆqr +ˆcq†rcˆqr) (26) r,q X or: cT = r(ˆaq†raˆqr +ˆcq†rcˆqr) (27) r,q X In a similar way T 0 and T i as components of a four vector are part of a time- space tensor, T˜µν = iψγ¯ µ∂pν ψ (28) whose other components yield the space boosts of the system as in analogy with Eq.20 the time operator T µ can be shown to be the generator of energy- momentum displacements , i.e.:

µ µ iqµT iqµT e Φ(p)e− = Φ(p + q) (29)

4 Interpretation and Conclusion

The inclusion of the self adjoint time operator and associated representation restores in QM the equal footing of time and energy accorded by SR, circumventing Pauli’s objection and providing an extended insight on the problem of time. The time space representation exhibits a total energy which is the sum of positive energy quanta for both particles and antiparticles present (Eq.17). Particles with negative energies are interpreted as antiparticles with positive energies. On the other hand, the energy-momentum representation exhibits a total intrinsic time as the sum of positive time quanta for both particles and antiparticles (Eq.25), giving a formal support to Feynman’s extraordinary identiﬁcation of the negative energy solutions evolving backward as positrons evolving forward in time. Second quantization, also referred to as occupation number representation, is a formulation that allows to describe states with varying numbers of particles

7 either free or bound by an external or a self consistent Hartree-Fock potential arising from their interactions. In the each case the energy quanta are the single particle energies, which comprise the discrete and continuum energy spectra in the single particle potential ei,e(p) . Then Eq.17 represent the total energy of the independent particle approximation,{ } where the energy quanta are the single particle energies. In a description where the ground state is the full Fermi sea, this energy accounts for the number of paticle-hole (electrons and positrons or protons and neutrons) states that define an excited state of the system. In the same way, Eq.25 represents a total intrinsic time summing up the corresponding individual time quanta expectation values, which are τ 0 for bound single particle states6. Thus the intrinsic time increases with the± number of particle-hole pairs involved in an excited state of the system. An early connection is found in Feshbach’s unified theory of nuclear reactions[25, 26] where sharp energy resonances are shown to be due to compound nucleus states. These are tipically many particle-hole configurations of mainly single particle bound states in the common potential. Unbound single particle states connecting to the open reaction channels will appear with very small ampli- tudes, leading to long delay times and long lifetimes[26, 28, 27], wheras in the present formulation they are shown to carry large intrinsic times. To conclude, the purpose of this paper stresses the extension of basic RQM with the existance a self-ajoint time operator and the additional basis it provides. Some applications have been allready identified. Observable effects in experiments that simulate the Dirac equation[22] and in tunneling in attosecond optical ionization[23] have allready been associated with the existence of the time operator in the first quantization of SR. The time operator also provides support for the conditional interpretation of time in QG[24]. However its full impact in the extensive development and applications of QFT remains to be explored, both theoretically and experimentally. To be noted is that Feshbach resonances are currently subject of extensive reseach in the fields of cold atom systems[29] (where they provide the essential tool to control the interaction between atoms), and of Bose-Einstein condensates[30, 31]. The many facets of the problem of time in QG[2, 3] are also to be considered in the present context.

6In the RQM description of atom and nuclear bound states appear as a consequence of attractive potentials. If these depend solely on position, e.g., the Coulomb potential in atoms, the shell model self consistent potential in nuclei, the time operator T = α.r/c + βτ 0 satisﬁes the same commutation relation as with the free particle Hamiltonian[9], The expectation value of T in a bound state n,.is:

hT in = hn | T | ni = hn | α.r/c + βτ 0 | ni = ±τ 0 as the ﬁrst term is zero, α being a non diagonal matrix.

8 5 Appendix A Time operator eigenvalues and eigenvectors

ipx/~ i p0x0 p.r//~ A plane wave solution of the form Ψ(r, x0)= e− ψ(r)= e− { − }ψ(r) in Eq.8 yields the eigenvalue equation: ∂ i~cαi + βm c2 ψ(r)= eψ(r) (A.1) {− ∂xi 0 } with positive and negative eigenvalues:

e = (cp)2 + (m c2)2 (A.2) ± 0 and eigenvectors (”energy” spinorsp ):

eq = u (p) p e>m c2 and eq = w (p) p e< m c2 (A.3) | i q | i 0 | i q | i − 0 − where

2 1/2 ep + m0c χq σ p uq(p) = 2 c . 2m c 2 χq 0 ep+m0c 2 1/2 cσ.p ep + m0c 2 χ w (p) = ep+m0c q (A.4) q − 2m c2 χ 0 q 1 where e = + (cp)2 + (m c2)2 and for q = 1, 2 , χ = and χ = p 0 − 1 0 2 0 p . Note that the negative energy spinors have opposite spin projection to 1 the corresponding positive energy spinors. (”Given this deﬁnition, the charge conjugation operation C transforms the spinors uq(p) into wq(p) and viceversa”).

For a particle in an attractive r dependent potential with bound states, these spinors satisfy the equation −

i ∂ 2 i~cα + V (r)+ βm0c ψ(r)= eψ(r) {− ∂xi } q q q where e = Ei, E , the Ei being the bound states ϕi (r) eigenvalues and E the energy{ } continuum{ } of distorted eigenfunctions. The expectation value of the time operator in a bound state is then

q q q q T = drϕ ∗(r) α.r/c + βτ ϕ (r)= drϕ ∗(r)βτ ϕ (r)= τ h ii i { 0} i i 0} i ± 0 Z Z as α.is not a diagonal operator in spin space. ipx/ In the same way, a plane wave solution of the form Φ(p,p0)= e− φ(p)= i pˆ0x0 p.x//~ e− { − }φ(p) in Eq.15 yields the eigenvalue equation:

i ∂ i(~/c)α + βτ 0 φ(p)= tφ(p) (A.5) { ∂pi }

9 with positive and negative eigenvalues:

t = (r/c)2 + (τ )2 (A.6) ± 0 and eigenvectors (”time” spinorsp):

tq = u(r) r t > τ and tq = w(r) r t< τ (A.7) | i q | i 0 | i | i − 0 with

1/2 tr + τ 0 χq uq(r) = σ.r/c 2τ 0 χ tr +τ 0 q ! 1/2 σ.r/c tr + τ 0 χ w (r) = tr +τ 0 q (A.8) q − 2τ χ 0 q ! 1 0 where t =+ (r/c)2 + (τ )2 and for q =1, 2 , χ = , χ = . r 0 1 0 2 1 As eigenvectorsp of self-adjoint energy and time operators, energy and time spinors constitute complete orthogonal sets eq , tq as: (eq. 6.3.15 ) {| i} {| i}

e e e e u¯q(p)us(p) = δqs u¯q(p)ws(p)=0 w¯e(p)¯wee(p) = δ w¯e(p)ue(p) = 0 (A.9) q s − qs q s and similar for the time spinors. They both provide representations for a system state vector.

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