arXiv:2010.09136v1 [quant-ph] 18 Oct 2020 · 2020. 10. 20. · berg equation for A~y(0). This de nition is in accordance with the proposal in [7] which originated from relativis-tic

Space-time Quantum Actions

N. L. Diaz,1 J. M. Matera,1 and R. Rossignoli1, 2

1Departamento de F́ısica-IFLP/CONICET, Universidad Nacional de La Plata, C.C. 67, La Plata (1900), Argentina2Comisión de Investigaciones Cient́ıficas (CIC), La Plata (1900), Argentina

We propose a formulation of quantum mechanics in an extended Fock space in which a tensorproduct structure is applied to time. Subspaces of histories consistent with the dynamics of aparticular theory are defined by a direct quantum generalization of the corresponding classical action.The diagonalization of such quantum actions enables us to recover the predictions of conventionalquantum mechanics and reveals an extended unitary equivalence between all physical theories.Quantum correlations and coherent effects across time and between distinct theories acquire arigorous meaning, which is encoded in the rich temporal structure of physical states. Connectionswith modern relativistic schemes and the path integral formulation also emerge.

I. INTRODUCTION

Quantum mechanics (QM) is a mathematical frame-work for the development of physical theories [1]. Thisframework assigns an operator acting on a Hilbert spacefor each observable of a given system, e.g. the position ofa particle. In particular, the Hamiltonian operator cor-responds to the energy of the system and determines itsquantum evolution, defining thus the particular theory.On the other hand, the spectral properties of a generalHamiltonian preclude the introduction of a time oper-ator, a result known as Pauli’s theorem [2–4]: In thecanonical formulation of QM time is treated “clasically”,i.e. it is not part of the framework as an observable [3].

This manifest asymmetry between space and time is inclear contrast with the covariance of classical (relativis-tic) physics, a problem partially overcome in canonicalformulations of relativistic quantum field theories: Clas-sical theories are quantized on a time-slice [5] and spacebecomes an index indicating the site of an “oscillator”.In this way, transformations mixing space and time (e.g.Lorentz transformations) can be introduced. However,since the latter is an external parameter, not the indexof a site, at the Hilbert space level an asymmetry is stillpresent [6, 7]: A tensor product structure is applied tospace but not to time, as observed in [8–14]. This is amanifestation of fundamental open problems concerningthe proper treatment of general covariance on Hilbertspace [8, 15–19], which are an important motivation forthe recent interest on the introduction of time in a purelyquantum framework [3, 6, 7, 10–16, 20–27]. However, theasymmetry is present in any composite system [10, 11].In particular, this prevents the representation of trajec-tories in a Hilbert space (see Sec. II A) and the use ofconventional tools for describing quantum correlations intime [28, 29].

In this work, the conventional framework of QM is gen-eralized to remove the above-stated asymmetry. This isaccomplished by formulating quantum mechanics in anextended Fock space in which a tensor product structureis applied to time (previous attempts in this directioninclude [9], see discussion in Sec. II A). The formalismis presented in section II, together with the concept of

space-time quantum actions and the definition of physicalstates. The case of quadratic theories is analyzed in detailin section III, where connections with other formalismsthrough second quantization and relativistic considera-tions are also examined. Different proposals for obtain-ing physical predictions in the general case within thepresent extended framework, including states at a giventime through quantum foliation and path integrals, arediscussed in section IV. A final discussion is provided insection V.

II. FORMALISM

A. A Hilbert Space for Quantum Trajectories

We introduce in this section a Hilbert space H suitedfor representing trajectories (see Fig. 1) of a set of bosons

defined by operators ai, a†j , [ai, a

†j ] = δij , [ai, aj ] = 0,

for i, j arbitrary quantum numbers (e.g. i may repre-sent a discretized position x), which generate a “conven-

tional” Fock space H of states∏i(a†i )ni |0〉 (with ai|0〉 =

0). For this purpose we define creation/annihilation

operators Ai(t), A†j(t) on “each” time-slice, satisfying

[Ai(t), Aj(t′)] = 0 and

[Ai(t), A†j(t′)] = δ(t− t′)δij , (1)

with Ai(t)|Ω〉 = 0 ∀t ∈ [−T/2, T/2], which generate anextended Fock space H. Here |Ω〉 =

⊗j |0〉tj , where

the tensor product is to be interpreted as the continuumlimit of equally spaced discrete time “sites” with spacing�, such that tj = �j, j ∈ Z and Ai(tj) = Aitj/

√�, with

Aitj |0〉tj = 0 and [Aitj , A†i′tj′

] = δjj′δii′ . The algebra of

Eq. (1) is recovered from δ(tj − tj′) ≡ δjj′/�.The extended Hilbert space H of states∏i,j(A

†itj

)nij |Ω〉 can then be written as H =⊗

j Htj

with Htj the Fock space generated by the operators A†itj

(fixed j). Note also that we can write H =⊗

i Hi andthen

H =⊗i,j

Hij ,

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with Hij ≡ Hitj , which is the aimed Hilbert space sym-metry between “space” (index i) and time (see Fig. 1).

FIG. 1. Representation of two classical (distinguishable) par-ticles moving in flat space-time whose trajectories can be pa-rameterized as (t, qa(t), qb(t)) (top left). Conventional QMdescribes this situation by employing a basis of product states|q〉 = |qa〉⊗ |qb〉 which represent the positions at a given timein the Hilbert space H. Instead, in H the whole paths are rep-resented by |q(t)〉 = |qa(t)〉 ⊗ |qb(t)〉 ∝

⊗j |qatj 〉 ⊗ |qbtj 〉 (Eq.

(7)), where |qi(t)〉 ∝⊗

j |qitj 〉 (top-right) which establishesa completely symmetric application of the tensor product tospatial and temporal degrees of freedom. Moreover, classicaltime evolution q(t) → q(t + ∆t) can be seen from a passivepoint of view as a displacement t→ t−∆t of the whole man-ifold. In our formulation, quantum time evolution emergesfrom eiPt(−∆t)|q(t)〉 = |q(t + ∆t)〉. The symmetry betweenspace and time is further depicted on the bottom panel witha different example: The tensor product in space of a conven-tional quantum field theory is here extended to space-time.

This construction allows us to specify, up to quantumuncertainty, a classical trajectory in phase space as a co-herent history state, i.e. a product state of the form

|α(t)〉 := exp[∫

dtα(t) ·A†(t)]|Ω〉 , (2)

where α(t) · A†(t) =∑i αi(t)A

†i (t) (or an integral for

continuum labels).Here exp

[∫dtO(t)

]=⊗

j exp [�O(tj)], where O(t) ≡O(A(t),A†(t), t), such that

A(t)|α(t)〉 = α(t)|α(t)〉 . (3)

Note that |α(t)〉 = eαA† |Ω〉, whereA† =∫dtα(t)·A†(t)/α

with α = [∫dt|α(t)|2]1/2, is a “collective” trajectory bo-

son creation operator. The (over)complete set of thesetrajectories span H:

∫D2α(t) e−

∫dt |α(t)|2 |α(t)〉〈α(t)| = 1 (4)

where D2α(t) :=∏i,j

d2αi(tj)π �.

Alternative basis are provided for example by opera-

tors Q(t) = A(t)+A†(t)√

2, P (t) = A(t)−A

†(t)

i√

2, such that

[Qi(t), Pj(t′)] = iδ(t− t′)δij (5)

(we set ~ = 1). Then we can define the correspondingeigenstates |q(t)〉, |p(t)〉, satisfying

Q(t)|q(t)〉 = q(t)|q(t)〉 , P (t)|p(t)〉 = p(t)|p(t)〉 . (6)

Explicitly, we can write [30], [31, 32]

|q(t)〉 = exp[− 12∫dtA†(t) · (A†(t)− 2

√2q(t))]|Ω〉 (7)

such that |q(t)〉 =⊗

j γj |qtj 〉tj with qtj =√�q(tj), γj =

4√π e|qtj |

2/2 and tj 〈qtj |q′tj 〉tj = δ(q − q′). The complete-

ness relation reads∫Dq(t) e−

∫dt |q(t)|2 |q(t)〉〈q(t)| = 1

(Dq(t) =∏i,j dqi(tj)

√π�). Similar formulas hold for

|p(t)〉. These space-time bases enable a novel approachfor path integral representations, as will be discussed inSec. IV B.

While H is isomorphic to a tensor product of copies intime of H, we have not specified any particular time evo-lution yet. We have only introduced a suitable “geomet-rical” scenario (which may be indicated as space-time) inwhich any laws of physics may be defined. In fact, a ketin H does not “evolve” but it can contain by itself all thetime information (or history) of a given system. Somecondition must establish which ones of these histories iscompatible with a particular theory, an intuition whichleads us to the definition of physical subspaces HP . Itturns out that if we propose that the trivial theory (nullHamiltonian) is defined by those coherent states invari-ant under time translations, a natural definition for alltheories follows. This result, which is presented in Sec.II C, relies on the extended unitary equivalence betweentheories that we introduce in Sec. II B.

We also note that a similar discrete tensor productin time Hilbert space is employed in the context of the‘consistent-histories’ approach to quantum mechanics in-troduced by Isham [8], with the aim of providing a novelway of representing the corresponding decoherence func-tional. The latter is the central quantity in the schemedeveloped in [33, 34], concerning the joint probability offinding a sequence of properties at a series of times. InIsham’s approach, a copy of the original Hilbert spaceis involved for each of these times. In its continuous-time formulation [9], the basic operators also satisfy Eq.

3

(1). Nevertheless, in the present formalism, this enlargedHilbert space, rather than a tool for representing histo-ries, is considered as fundamental. In particular, timeevolution is derived from properties of the correspondingtime translation operator and encoded in physical stateswhile the “number of time sites” is arbitrary. Quanti-ties such as the decoherence functional can be obtaineda posteriori.

B. Time Translations and Space-time QuantumActions

Consider the generator of time translations Pt in thepresent scenario, defined as

Pt :=∫dω ωA†(ω) ·A(ω) (8a)

=

∫dtA†(t) · iȦ(t) (8b)

= 12

∫dt [P (t) · Q̇(t)−Q(t) · Ṗ (t)] (8c)

where A(ω) is the Fourier transform (FT) of A(t), suchthat A(t) =

∫dω√2πA(ω)e−iωt (continuous notation, see

Appendix A) and iȦ(t) =∫

dω√2πA(ω)ωe−iωt coincides

with the “site” derivative (Eq. (A3)). We assume peri-odic conditions A(−T/2) = A(T/2). The operator Ptsatisfies

eiPt∆tA(t)e−iPt∆t = A(t+ ∆t) , (9)

which for ∆t→ 0 leads to

[Pt,A(t)] = −iȦ(t) , (10)

in agreement with Eq. (8b).Remarkably, the integrand in (8c) has the form of the

Legendre transformation which connects the Hamiltonianwith the Lagrangian in classical mechanics. This suggeststhe introduction of a new object that for the trivial theoryreduces to Pt:

J :=∫dt [A†(t) · iȦ(t)−H(A(t),A†(t), t) , (11)

which will be indicated as space-time quantum actionoperator (not to be confused with Schwinger’s action[35], [36]) for its formal coincidence with the classical

one. Here∫dtH(A(t),A†(t), t) ≡

∑tH(At,A

†t , t) for

H(a,a†, t) a conventional (quantum) Hamiltonian (anddt = �), in accordance with the convention of J havingunits of Pt. A remarkable result is that J and Pt areunitarily related (see proof in Appendix B):

J = V†PtV =∫dω ωÆ(ω) · Ã(ω) (12a)

=

∫dt Æ(t) · i ˙̃A(t) (12b)

= 12

∫dt [P̃ (t) · ˙̃Q(t)− Q̃(t) · ˙̃P (t)] , (12c)

where

V† := T̂ ′ exp[−i∫dt

∫ tt0

dt′H(A(t),A†(t), t′)]

(13)

is a tensor product in time of conventional time evolution

operators U(t, t0) = T̂′ exp[−i

∫ tt0dt′H(a,a†, t′)] (T̂ ′ de-

notes time ordering applied to t′) and

Ã(ω) = V†A(ω)V , Ã(t) = V†A(t)V , (14)

with Ã(t) the FT of Ã(ω) (similarly Q̃(t) = V†Q(t)V,P̃ (t) = V†P (t)V). Here t0 is a reference time such thatÃ(t0) = A(t0). In particular, for H time independent,

V† = exp[−i∫dt (t− t0)H(A(t),A†(t))] . (15)

Since in this context J is the operator that definesa particular time evolution (Sec. II C), the result (12a)is unitarily relating all theories to the trivial one. Thisalso means that in H all physical theories appear unitar-ily related between themselves. Such general result is aconsequence of the remarkable property of the space-timequantum actions of having the same spectra regardless ofthe Hamiltonian. This should be compared with the ob-vious fact that different Hamiltonians have different spec-tra, which also means that such unitary relation betweentheories could have never been revealed in a Hamiltonianformulation.

The proof of (12a) is based on the basic properties ofPt as the generator of time translations, and assumesperiodic conditions for finite T (something which in prin-ciple can always be “enforced” or implemented by a “wellbehaved” H in the limit T → ∞). Notice that Eqs. (9)and (12) entail

eiJ∆tÃ(t)e−iJ∆t = Ã(t+ ∆t) (16)

such that J is the generator of time translations in the“normal” basis for a non-null Hamiltonian. Therefore,the operators Ã(t) satisfy

[J , Ã(t)] = −i ˙̃A(t) , (17)

in accordance with (12b). In fact, they are the unique

annihilation operators fulfilling (17) and Ã(t0) = A(t0).The uniqueness is an immediate consequence of (16)which implies

Ã(t) = eiJ∆tA(t0)e−iJ∆t (18)

when ∆t = t − t0. The relation (18) is a remarkable re-sult on its own which provides an expansion in powersof ∆t of the “evolved” operator V†A(t)V (see also Ap-pendix B and the discussion below). In the context of theconsistent histories approach, and for the particular caseof a time-independent harmonic oscillator, an analogousaction complying with Eq. (18) was introduced in [37].

4

Before proceeding to the definition of physical sub-spaces, we would like to stress that as a consequenceof (12)–(14) the information of conventional time evo-

lution is already encoded in the operators Ã(t): From

Eq. (13) it is clear that the operator Ã(t) correspondsto the operator a(t) = U(t, t0)aU

†(t, t0), which acts onHt. Since an underlying tensor product is involved, thisstatement is rigorous for discrete time, in which case wecan also speak properly of “instants” and “sites”. In Sec.IV these ideas and the discrete regularization will be em-ployed to derive (and interpret) different ways to obtainphysical predictions from the inner product of H. On theother hand, the expressions involved can also be obtainedstraightforwardly in the ω basis by employing the normaloperators Ã(ω) of (12a), which satisfy

[J , Æ(ω)] = ωÆ(ω) . (19)

In this basis the limit � → 0+ is well defined and a mapwith conventional states in H can be easily introduced.

We also remark that for a general periodic (orwell behaved in the limit T → ∞) operator U =exp[

∫dtM(A(t),A†(t), t)] , Eq. (9) yields (see Appendix

B for the details)

[Pt,U ] = i∂U∂t

(20)

with ∂U∂t defined in (B7) through Eq. (10). For

M(A(t),A†(t)) time independent, Eq. (B7) implies[Pt,U ] = 0. If iM(A(t),A†(t)) is also hermitian, thisimplies U†PtU = Pt, i.e. Pt is invariant under time in-dependent canonical transformations A(t) → U†A(t)U .This means that without imposing any initial conditions,the diagonal form (12a) is not unique and implies

[U ,∫dtH(A(t),A†(t), t)] = 0⇒ [U ,J ] = 0 . (21)

In particular, a time-independent symmetry of H,[M(a,a†), H(t)] = 0, is a symmetry of J : [U ,J ] = 0, for�M(A(t),A†(t)) = M(At,A†t). On the other hand, forH time-independent it follows from Eq. (9) that eiPt∆t

satisfies Eq. (21), i.e. J is invariant under time transla-tions and hence [Pt,J ] = 0 (see also Eq. (B9)). In theAppendix B we discuss further symmetries of Pt and Jwhich are not diagonal in time, together with the pos-sibility to generalize (12a) to “exotic” theories involvingmultiple-times.

Finally, it is appropriate to mention that differentdefinitions of time localization are now possible: As ithappens for spatial localization in quantum field theo-ries (QFT) with important implications on spatial un-certainty relations [38, 39], time localization is now anemergent aspect of the “lattice”. Different definitions ofthis notion would also imply different energy-time uncer-tainty relations according to the operators involved. Anexample is provided by the single particle (sp) time oper-ator T :=

∫dt tA†(t) ·A(t) which reduces on sp states to

the Page and Wootters (PaW) operator [40] (see Sec. IVD) employed in other recent formalisms with quantumtime [3, 6, 7, 15, 20, 22–25]. In this case, it can be shownthat (see Eq. (B7); here T →∞)

[Pt, T ] = iN , (22)

where N :=∫dtA†(t) ·A(t) =

∫dωA†(ω) ·A(ω) is the

number operator (e.g. N (A†i (t))ni |Ω〉 = ni(A†i (t))

ni |Ω〉).Then ∆T ∆Pt ≥ 12 |〈N 〉| through the Cauchy–Schwarzinequality in H. Despite the importance of the energy-time pair in QM [4], this treatment is usually preventedby the impossibility of introducing a time operator in H[2, 4, 41].

C. Physical States

We are now in a position to formalize the postu-lates that define a particular physical theory: Considerthe normal operators Ã(ω) defined by the representa-tion (12a) of the quantum action, fulfilling Eq. (19) and

Ã(t0) = A(t0), and their vacuum |Ω̃〉 = V†|Ω〉. The cor-responding HP is introduced as the linear space spannedby states

∏i(Æi (ω = 0))

ni |Ω̃〉, i.e. the Fock space gener-ated by the creation operators satisfying

[J , Æ(0)] = 0 , (23)

which may be interpreted as a static (or timeless) Heisen-

berg equation for Æ(0). This definition is in accordancewith the proposal in [7] which originated from relativisticconsiderations. In particular, since just ω = 0 bosons areinvolved, J |Ψ〉 = 0 ∀ |Ψ〉 ∈ HP , a constraint which de-fines related quantum formalisms [40, 42] motivated bythe Wheeler-DeWitt equation [43] (see also Sec. III D).Eqs. (12a), (23) also imply 〈Ψ| δJ

δÃ(ω)|Ψ〉 = 0, meaning

that the average of the quantum action J is stationaryin HP as a functional of Ã(ω) [44].

In order to show that the present formalism yields (ina physical subspace) the same predictions of conventionalQM, we establish an isomorphism L : H→ HP such that

L(∏

i

[(a†i )ni ]|0〉

)=∏i

[(Æi (0))ni ]|Ω̃〉 . (24)

We will say that |Ψ〉 = L(|ψ〉) is the history of |ψ〉 ∈ Hwith the Hamiltonian that defines J . In particular, fora coherent state |ψ〉 = eα·a† |0〉, (24) leads to

|Ψ〉 = exp[α · Æ(ω = 0)]|Ω̃〉 = exp[∫

dt√Tα · Æ(t)

]|Ω̃〉

= V† exp[∫

dt√Tα ·A†(t)

]|Ω〉 , (25)

which is a product of evolved states when V† is the oper-ator (13). Thus, the time invariance proposed for history

5

coherent states of the trivial theory (H = 0, V† = 1) uni-tarily defines any other. An important property followsfrom (24): If |Φ〉 is the history of |ϕ〉 then

〈Φ|Ψ〉 = (L(|ϕ〉),L(|ψ〉)) = 〈ϕ|ψ〉 , (26)

and in particular 〈Ψ|Ψ〉 = 〈ψ|ψ〉, a relation which holdsfor any T , as it follows from [Ãi(0), Ã

†j(0)] = δij . More-

over, even if an infinite extent of time is considered, anatural approach emerges: The formalism treats ω asa usual continuous quantum number with an associatedeigenfunction expansion. This may be regarded as aneigenbasis associated with different physical theories la-beled by ω: A state can be normalized if a quantum un-certainty in the physical theory is allowed (see App. C).

III. THE QUADRATIC CASE

A. Quadratic Space-time Quantum Actions

In the following, we explicitly develop the case ofbosonic quadratic theories as an important example of(11). For a general quadratic Hamiltonian [45],[46]

H(a,a†) = 12(a† a

)(ω0(t) γ(t)γ∗(t) ω∗0(t)

)(aa†

)= 12ψ

†K(t)ψ

where ω0 (γ) are hermitian (symmetric) matrices andψ = (aa†) satisfies

Π = [ψ,ψ†] := ψψ† − ((ψ†)tψt)t =(1 00 −1

),

the quantum action (11) becomes

J = 12∫dt[Ψ†(t)ΠiΨ̇(t)−Ψ†(t)K(t)Ψ(t)] , (27)

with Ψ(t) = (A(t),A†(t))t, [Ψ(t),Ψ†(t′)] = Πδ(t−t′). Itis first verified that under any constant Bogoliubov trans-

formation (BT) Ψ(t) → W0Ψ(t), where W †0 ΠW0 = Π(linear time independent canonical transformation), the

form of J is preserved (with K → W †0KW0). It is thenseen that the diagonal form (12a)

J = 12∫dt Ψ̃†(t)Πi

˙̃Ψ(t) = 12

∫dω ω Ψ̃†(ω)Ψ̃(ω) , (28)

can be achieved by applying in (27) a diagonal in timeBT

Ψ(t) = W (t)Ψ̃(t) , (29)

where W (t) satisfies the Heisenberg equation [47]

iẆ (t) = ΠK(t)W (t) (30)

with W (t0) = 1 in order that Ψ̃(t0) = Ψ(t0) (im-plying W †(t)ΠW (t) = Π ∀t). This is in agreement

with Eqs. (13)–(14) since in the present case V =exp[ i2

∫dtΨ†(t)M(t)Ψ(t)] with e−iΠM(t) = W (t), and

V†Ψ(t)V = Ψ̃(t), V†Ψ(ω)V = Ψ̃(ω) (31)

are BTs equivalent to (29).This is the only solution satisfying the initial condition

Ã(t0) = A(t0), as we proved in Eq. (18).

B. Time Structure of Physical States

It is important to remark that the states |Ψ〉 ∈ HPconstructed with Eq. (24) already contain all time infor-mation of the system, in a nontrivial way. In fact, generalphysical states |Ψ〉 = L(|ψ〉) have a complex time struc-ture and in particular exhibit in general entanglementin time, even for decoupled oscillators: By considering

H =∑i ω

i0(a†iai +

12 ) [48] Eq. (27) becomes

J =∑i

∫dω (ω − ωi0) (A

†i (ω)Ai(ω) +

12 ) (32)

such that Ãi(ω) = Ai(ω + ωi0) in (12a) and Ãi(t) =

eiωi0tAi(t), in agreement with (29)–(30). Then a sp state

Æi (ω = 0)|Ω〉 =∫

dt√Teiw

i0tA†i (t)|Ω〉 =

∫dt√Teiw

i0t|ti〉

(33)is an W -like state in the time representation (unlocal-

ized in time), where we have written |ti〉 = A†i (t)|Ω〉.A general sp physical state then has the formal ap-pearence of a PaW state [3, 22] (see also Sec. III D)

|Ψ〉 =∫

dt√T

∑i ψie

iwi0t|ti〉. However, more general Fockstates, e.g.

(Æi (0))2|Ω〉 =

∫dt1√T

dt2√Teiw

i0t1eiw

i0t2A†i (t1)A

†i (t2)|Ω〉 ,

(34)have even a richer structure.

On the other hand, an initial coherent state leads tocoherent product state (Eqs. (2) and (25))

L (|α〉) = |α(t)〉 =⊗i,j

exp

[αie

iωi0tj√T/�

A†itj

]|Ω〉 (35)

i.e. (α(t))i =(α)i√Teiw

i0t, implying

L

(∫ ∏i

d2αiπ ψ(α)|α〉

)=

∫ ∏i

d2αiπ ψ(α)|α(t)〉 . (36)

We conclude that the physical subspace of time-independent stable quadratic systems corresponds to thelinear space of quantum trajectories |α(t)〉, where α(t)is a solution of the classical equations of motion. These“almost” classical trajectories also have a “classical time

6

structure”, namely separability in time, which is an ap-pealing property. Remarkably, L(|ψ〉) has the same for-mal expansion of |ψ〉 in this basis, although notice thatsuch superposition of separable (but composite) stateswill in general be entangled.

C. Physical Predictions

Physical operators defined by Eq. (31) satisfy, for Ktime independent (∆t = t− t0)

eiPttΨ̃(0)e−iPt∆t = exp(−iΠK∆t)Ψ̃(0) , (37)

where Ψ̃(0) = Ψ̃(ω = 0). This result is to be comparedwith the standard Heisenberg operators for the quadraticcase,

eiH∆tψe−iH∆t = exp(−iΠK∆t)ψ

and has a clear geometrical meaning: a rigid translationof the time sites reproduces the conventional time evolu-tion of physical operators. The details can be found inAppendix D. This result also holds in the time-dependentcase by replacing eiPt∆t with the unitary “complete”time-translation operator W(∆t) from Eq. (E1) whichtranslates both the time sites and the explicit time de-pendence of H such that [W(∆t),J ] = 0 (see AppendixE).

From Eq. (37) it follows that if O(t) =eiPt∆tO(Ψ̃(0))e−iPt∆t for O an arbitrary function

of Ψ̃(0), then

〈Φ|O(t)|Ψ〉 = 〈ϕ|OH(t)|ψ〉 (38)

for OH(t) = eiH∆tO(ψ)e−iH∆t and |Ψ〉 (|Φ〉) the history

of |ψ〉 (|ϕ〉), a relation which holds for any quadraticHamiltonian, observable and states. The generalizationto the time-dependent case and multiple-time correlationfunctions is apparent.

Moreover, time translations preserve the separation be-tween the ω = 0 mode and the rest, implying

〈Φ|e−iPt∆t|Ψ〉〈Ω̃|e−iPt∆t|Ω̃〉

=〈ϕ|e−iH(a,a†)∆t|ψ〉〈0|e−iH(a,a†)∆t|0〉

, (39)

which reduces to Eq. (26) for t = t0. An explicit deriva-tion of Eq. (39) is provided in the Appendix D, which alsoshows its invariance under linear symmetries of J (non-necessary diagonal in time). Its time-dependent versionis derived in Appendix E.

D. Second Quantization of Parameterized Particlesand PaW formalism

One important motivation of the present formulationwas to remove the asymmetry between “space” and time

in QM by incorporating the latter in the same frame-work. Different aspects of this problem are treated inthe quantization of reparameterization invariant systems[15, 42] and related quantum formalisms like the one pro-posed by Page and Wootters [40] (and recent revisions[3, 20, 22, 24, 25], including the relativistic extensions[6, 7] relevant for the present scheme). Here we dis-cuss how these other proposal are connected to our workthrough the sp space of particular spaces H.

The treatment of a parameterized particle (one di-mensional for simplicity) for a time independent La-grangian L(q, q̇) leads to a classical weak constraint [42]

HS = pt + H ≈ 0 with pt = ∂(ṫL(q,q̇/ṫ))∂ṫ . This conditionis quantized as [49, 50]

HS |Ψ〉 = (Pt ⊗ 1 + 1⊗H)|Ψ〉 = 0 , (40)

where Pt ⊗ 1 = i∫dtdt′dq ddt′ δ(t

′ − t)|tq〉〈t′q|, 1 ⊗ H =∫dtdqdq′ 〈q′|H|q〉|tq′〉〈tq| and

〈t′q′|tq〉 = δ(t− t′)δ(q − q′) , (41)

which is commonly considered as an auxiliary conditionon a “kinematic space” K to define the physical space(which is not a proper subspace). Alternatively, a rela-tional interpretation is assigned to this equation whereHS is regarded as the Hamiltonian of a composite globalsystem “clock”+“system”. This is the case of the PaWformalism where an hermitian time operator is definedas the observable of the clock T =

∫dt t|tq〉〈tq|.

If instead the kinematic space is promoted to the sta-tus of a “physical” space and, moreover, the particles areregarded as a d + 1 dimensional objects (for d spatialdimensions), the proper scenario for many identical par-ticles is an extended Fock space H [7], different from theconventional one and different from the PaW formalismapplied to a Fock space (or equivalently, from the general-ized Hamiltonian dynamics of a conventional Fock space).This is achieved by reinterpreting the states |tq〉 as spstates |tq〉 = A†(t, q)|Ω〉 (with A(t, q)|Ω〉 = 0, 〈Ω|Ω〉 = 1)which, considering Eq. (41) and a bosonic particle, im-plies [A(t, q), A†(t′, q′)] = δ(t − t′)δ(q − q′), an exampleof (1). Then one may generalize

HS → −J (42)

with

J =∫dt

∫dqdq′A†(t, q′)[i∂tδ(q − q′)− 〈q′|H|q〉]A(t, q)

(43)which remarkably is the space-time quantum action (11)for a field of harmonic oscillators (here i → q) and asingle particle Hamiltonian (for a local H, J becomeslocal in space-time), a particular instance of the generalquadratic case (27). As a consequence, sp states (but notmultiparticle states) in H are formally identical to PaWstates while the sp matrix elements of the operators J , Tare equal to the matrix elements of HS , T respectively

7

(including J |Ψ〉 = 0 for |Ψ〉 ∈ Hp being formally equiv-alent to Eq. (40) for sp states). Notice however that theproduct structure between “time” and “rest”, essentialfor “conditioning on a clock”, is completely lost [7]: Theproduct structure of H is applied to time itself with ageometrical rather than relational meaning. As a conse-quence, our definition of foliation (of Sec. III D) works ona different basis without any reference to a clock.

FIG. 2. On the left, the two descriptions of the single parti-cle: The conventional one in the Hilbert space H (top panel)and the generalized description in space-time in the HilbertK (bottom panel). On the right, the second quantization ofthe previous schemes. The second quantization of H leadsto a field theory in a conventional Hilbert space HF which isisomorphic to a tensor product in space of copies of H, i.e.HF ≈

⊗q Hq (top right panel). The second quantization of

K leads instead to an extended space H ≈⊗

t HFt =

⊗t,q Htq

where the tensor product structure is applied to both spaceand time and it is possible to represent field configurationsin space-time (bottom right panel). The description of thefield in this extended Hilbert space can be immediately ob-tained by applying the formalism presented in this work tothis particular case.

Note also that the second quantization [51] of theconventional Hilbert space H of the particle, which isspanned by states |q〉, leads as well to a field theory, nowin a Fock space HF generated by operators a†(q) suchthat |q〉 = a†(q)|0〉. This is the system described in thepresent Hilbert H: J in Eq. (43) is precisely the space-time quantum action which corresponds to the Hamilto-nian

H =

∫dqdq′〈q′|H|q〉a†(q′)a(q) (44)

obtained through second quantization of the Hamilto-nian of the particle. The relation between these differentHilbert spaces is represented in Fig. 2. An independent

description of the particle (without the field) can be pro-vided in a different H for H the Hamiltonian of the par-ticle in Eq. (11).

We remark finally, that while in HF the product struc-ture applied to space allows to represent field configura-tions at a given time as eigenstates [30]

|φ(q)〉 = exp[− 12∫dq [a†(q)(a†(q)− 2

√2φ(q))]|0〉 (45)

of φ(q) = a(q)+a†(q)√

2, in H the product structure is ex-

tended to time allowing to represent space-time configu-rations

|φ(q, t)〉 = exp[− 12∫dtdq[A†(t, q)(A†(t, q)−2

√2φ(t, q))]|Ω〉 ,

(46)i.e. Eq. (7) applied to the present case.

E. Relativistic Considerations

The relativistic case was traditionally considered as aspecial case of non-relativistic QM [5] since, e.g. scalarfield theories can be interpreted as the continuum limit ofcoupled harmonic oscillators in space, an example of (27)for free theories. On other hand, the present formalismis particularly suited for a geometrical interpretation ofthe space-time sites: For i → x and Ai(t) → A(x), wedefine U(Λ) by U†(Λ)A(x)U(Λ) = A(Λx) (for T → ∞).The algebra implied by Eq. (1),

[A(x), A†(y)] = δ(4)(x− y) (47)

is explicitly preserved when Λ is a Lorentz transforma-tion. This yields U(Λ)|φ(x)〉 = |φ(Λ−1x)〉 for the coher-ent field state

|φ(x)〉 = exp[∫d4xφ(x)A†(x)]|Ω〉 (48)

(α(t)→ φ(x) in (2)), which is the correct transformationproperty of a state representing a (scalar) field config-uration in space-time (a similar reasoning holds for thestates (46) for q → x).

The generator of time translations transforms asU†(Λ)PtU(Λ) = Λ µ0 Pµ with Pµ :=

∫d4xA†(x)i∂µA(x)

such that P0 = Pt. In particular, [U(Λ),Pt] = 0 onlyin the limit of Galilean transformations. In order to in-troduce invariant physical subspaces we can employ aprevious proposal by the authors [7] (more recently alsopresented in [52]) which consists of considering a sec-ond quantization version of the constraint Hrels |Ψ〉 :=(PµPµ − m20)|Ψ〉 = 0 (and P 0 > 0) where the hermi-tian operators Pµ satisfy [Xµ, Pν ] = iδ

µν with X

0 = Tthe PaW time operator [7]. The constraint Hrels |Ψ〉 = 0also arises from the treatment of reparameterization in-variant systems but considering now the classical actionS = −m0

∫dτ [49, 50]. This treatment leads to

HrelS → Jrel = −∫d4xA†(x)(∂2 +m20)A(x) (49)

8

such that [U(Λ),Jrel] = 0 and implying

〈φ(x)|Jrel|φ(x)〉〈φ(x)|φ(x)〉

= S[φ(x), φ∗(x)] (50)

where S[φ(x), φ∗(x)] = −∫d4xφ∗(x)(∂2+m20)φ(x) is the

classical action of a free scalar field (η00 = 1, c = 1).The result (50) is suggesting a deep connection betweenparticle-like techniques and a formulation of QFT in thisextended setting.

This new form of the quantum action also admitsa normal decomposition (analogous to (32)) such that[Jrel, A†(m2,p)] = (m2 − m20)A†(m2,p) implying ineach mass sector the three-dimensional invariant prod-uct [7]. As a consequence, the correct commutators be-tween physical field operators (the component of φ(x) ∝A(x)+A†(x) at fixed mass) also emerge [53]. In fact, thedefinition (23) of physical states corresponds in this caseto the mass-shell condition (see also [7]).

Note that we could have considered instead J =∫d4p (p0 − Epm)A†(p)A(p) which yields an equivalent

constraint for Epm =√p2 +m2. This J has the form

(11) for H =∫d3pEpma

†(p)a(p) with [a(p), a†(p′)] =

δ(3)(p − p′), i.e. H is the (diagonalized and normal-ordered) Hamiltonian of the free scalar field we wantto describe. While explicit Lorentz symmetry is lost,under e.g. a boost in the first direction such thatp0 → cosh ηp0 + sinh ηp1, U†(Λ)JU(Λ) = cosh ηJ andthe physical subspace remains invariant:

[J , Æ] = 0⇔ [U†(Λ)JU(Λ), Æ] = 0 .

We see that the possibility to represent space-time con-figurations of the fields opens the possibility to explic-itly preserve the symmetries of space-time (Lorentz co-variance in the previous example) at the Hilbert spacelevel and in particular in quantization processes. As afundamental consequence, the correct invariant productemerge in Hp from the (standard) global inner productof H in the case considered [7].

IV. RECOVERING PHYSICAL PREDICTIONSIN THE GENERAL CASE

A. Quantum Foliations

For nonquadratic theories Eq. (37) (and its time-dependent version) no longer holds for V diagonal in timeas defined in Eq. (13). However, even for such diagonalsolutions, there is still a simple scheme to extract infor-mation “at a given time” from |Ψ〉: We introduce a uni-tary quantum foliation operator defined as the shifted in-verse FT F̃†(t)Ã(ω)F̃(t) :=

√�Ã(t+ �Tω/2π) such that,

roughly speaking, F̃†(t)|Ψ〉 contains the state U(t, t0)|ψ〉at the site t. We can make this statement more precisefor discrete time in which case (see Appendix A)

F̃†(tj)Ã(ωk)F̃(tj) = Ãtj+k , (51)

implying

F̃†(t)∏i

[(Æi (ω = 0))ni ]|Ω̃〉 =

∏i

[(Æit)ni ]|Ω̃〉

= V†∏i

[(A†it)ni ]|Ω〉 (52)

when t = tj = �j. Hence, given |Ψ〉 = L(|ψ〉) ∈ HP , weobtain

F̃†(t)|Ψ〉 = |ψ(t)〉j⊗j′ 6=j

|0(tj′)〉tj′ (53)

for |ψ(t)〉 = U(t, t0)|ψ〉, |0(t)〉 = U(t, t0)|0〉 and where weused V† =

⊗t U(t, t0). The unitarity of F̃(t) reflects the

unitarity of time evolution:

〈Φ|F̃†(t)F̃(t)|Ψ〉 = 〈Φ|Ψ〉 = 〈ϕ|ψ〉 ∀t (54)

for |Φ〉 = L(|ϕ〉) and in agreement with (26).We see that we can recover the evolved state |ψ(t)〉

from |Ψ〉 by first applying the foliation operator and thentaking the partial trace over the Hilbert spaces of theother times. This defines a CPTP (completely positivetrace preserving) map [1] which in particular for t = t0provides a representation of L−1. On the other hand,there are straightforward ways to obtain physical predic-tions which employ the inner product of the global spaceH. In the following we present results in this direction.

1. Propagators

Consider again |Ψ〉 = L(|ψ〉) and |Φ〉 = L(|ϕ〉). FromEq. (53) it follows that

〈Φ|F̃(t0)e−iPt(t−t0)F̃†(t)|Ψ〉〈Ω̃|e−iPt(t−t0)|Ω̃〉

=〈ϕ|U(t, t0)|ψ〉〈0|U(t, t0)|0〉

(55)

with 〈ϕ|U(t, t0)|ψ〉 the standard propagator. Heree−iPt(t−t0) moves |ψ(t)〉 (and the remaining vacua) backto site t0 where it overlaps 〈ϕ|. The remaining overlapsbetween vacua cancel with those in the denominator. Fort = t0 (54) is recovered.

The result (55) can be easily written in terms of theoriginal operators A(ω = 0), A†(ω = 0) or A(t), A†(t).For time-independent H, where time translations are asymmetry ([Pt,J ] = 0) the following simple expressionscan be obtained (∆t = t− t0):

〈Φ|F̃(t0)e−iPt∆tF̃†(t)|Ψ〉〈Ω̃|e−iPt∆t|Ω̃〉

=0〈Φ|e−iH(A(0),A

†(0))∆t|Ψ〉0〈Ω|e−iH(A(0),A†(0))∆t|Ω〉

(56)

=0〈Φ|F(t)eiJ∆tF†(t0)|Ψ〉0

〈Ω|eiJ∆t|Ω〉,

(57)

where |Ψ〉0, |Φ〉0, F(t) are in the trivial basis (see Ap-pendix F for the proof). Clearly, Eq. (56) agrees with

9

Eq. (55) and its limit � → 0+ is well defined. Inthe quadratic case, this equation reduces to (39) since[F(t),

∫dtH(A(t),A†(t))] = 0 for H(A(t),A†(t)) =∑

i ωi0A†i (t)Ai(t). The generalization for a time-

dependent H relies on the replacement eiPt(t−t0) →W(t− t0) and is developed in the Appendix E.

2. Observables and Correlation Functions

For H time-independent, Eq. (16) allows us to write(see also Eq. (A6))

eiPt�Ãtje−iPt� = eiH�Ãtj+1e

−iH� (58)

with H the Hamiltonian as a function of operators Ãti+1 ,

Æti+1 . We see that under the action of time translations,

the operators Ãti not only are translated into the newHilbert, but they are also evolving (see Figure 3). Moregenerally, (58) implies

eiPt∆tO(Ãtj , Ætj )e−iPt∆t = eiH∆tO(Ãtj′ , Ã

†tj′

)e−iH∆t

(59)

with H ≡ H(Ãtj′ , Ætj′

) and ∆t = tj′ − tj .

FIG. 3. Under time translations through ∆t/� steps, the

operator Ãtj is displaced to site tj′ = tj + ∆t while evolvingan amount ∆t (left panel). Through insertion of operators atdifferent times and translations back to the Hilbert at t = 0multiple-time correlation functions are obtained (right panel).

We can employ this point of view to obtain correlationfunctions: Given a conventional operator O(a,a†), whichin the Heisenberg picture reads OH(t) = e

iHtOe−iHt (weset t0 = 0), from (59) we obtain

〈ϕ|OH(tj)|ψ〉 = 〈Φ(0)|eiPttjO(Ã−tj , Æ−tj )e

−iPttj |Ψ(0)〉

= 〈Φ|eiHtjO(Ã(0), Æ(0))e−iHtj |Ψ〉 , (60)

for |Ψ(0)〉 = F̃†(0)|Ψ〉, |Ψ〉 = L(|ψ〉), |Φ(0)〉 = F̃†(0)|Φ〉,|Φ〉 = L(|ϕ〉). In the last equality we have “extracted”the operators F̃(0) from |Ψ(0)〉, |Φ(0)〉, such that Ã(0) =Ã(ω = 0) and H ≡ H(Ã(0), Æ(0)). This is of coursethe expression which is obtained by applying the map Lto both the states |ψ〉, |ϕ〉 and the operator O.

The result (60) can be immediately generalized to com-pute multiple-time correlation functions by “inserting”

now operators at different times: If we define

Oi(tji) := eiPttjiOi(Ã−tji , Æ−tji

)e−iPttji

then

〈ϕ|∏i

OiH(tji)|ψ〉 = 〈Φ(0)|∏i

Oi(tji)|Ψ(0)〉 . (61)

The corresponding ω expansion is apparent and only in-volves physical operators (operators acting on Hp).

All these relations, starting from Eq. (58), can begeneralized to the time-dependent case by replacingeiPt∆t →W(∆t) from Appendix E. A similar procedurecan be employed for the mixed case and for the moregeneral decoherence functional [34].

B. Path Integrals from Quantum Trajectories

The space-time quantum actions, their unitary equiv-alence with Pt and the “trajectory” states (2)-(6) alsoenable a straightforward novel approach to path inte-grals (PIs), which provides an alternative way to computephysical predictions. In order to illustrate this point, wewill show first that a conventional product of time or-dered operators in H can be expressed in H as

T̂(O1H(t1)O

2H(t2) . . . O

nH(tn)

)= Trt 6=0

[eiJ �O

], (62)

where OiH(t) = U†(t, 0)OiU(t, 0), ti = �ji, and

O := O1(Atj1 ,A†tj1

) . . . On(Atjn ,A†tjn

) (63)

is a product operator in time with Oi on the slice Htji(and identities for j 6= ji). The time-ordering emergesnaturally from the ordering of the time-sites in H. Thisalso provides an alternative representation of the prod-uct of operators in (61) (when the times are ordered):∏iOi(tji) = Trt6=0

[eiJ �O

].

Proof: Note first that Eq. (62) is equivalent to

〈ϕ|T̂O1H(t1)O2H(t2) . . . OnH(tn)|ψ〉 = Tr[|ψ〉〈ϕ|eiJ �O

]∀

|ψ〉〈ϕ| ≡ |ψ〉〈ϕ|⊗

j 6=0 1j acting on H0. On the other

hand, from the result (12a), and the initial conditionV†At=0V = At=0 (implying [V, |ψ〉〈ϕ|] = 0),

Tr[|ψ〉〈ϕ|eiJ �O] = Tr[|ψ〉〈ϕ|eiPt�VOV†] (64)

with

VOV† = O1H(At1 ,A†t1 , t1) . . . O

nH(Atn ,A

†tn , tn) , (65)

i.e. VOV† is a tensor product of operators OiH(t), eachone evolved up to the corresponding time site value.We then note that a quantity 〈ϕ|O1O2 . . . On|ψ〉 can berewritten as (i = . . . , i−2, i−1, i1, i2, . . . )

〈ϕ|O1O2 . . . On|ψ〉 =∑i

〈ϕi1i2 . . . |eiPt�O|ψi1i2 . . . 〉

= Tr[|ψ〉〈ϕ|eiPt�O

](66)

10

with∑i |i〉〈i| = 1 and the operators appearing in the

inverse order of the time-sites (here, to comply withthe ordering on the left hand side of (66) we shouldchoose t1 > t2 > · · · > tn in the definition of O.With the time-ordering operator this is no longer re-quired.). The expression (66) relies on a basic re-lation between quadratic forms and tensors [54] (e.g.〈ϕ|O1O2|ψ〉 =

∑i〈ϕ|O1|i〉〈i|O2|ψ〉 =

∑i〈ϕi|eiPO2 ⊗

O1|ψi〉 for e−iP |ϕi〉 = |iϕ〉). In (66) the time transla-tion operator eiPt� ensures the correct indices ordering.

The validity of Eqs. (64)-(66) ∀ |ψ〉〈ϕ| implies (62),with the time-ordering linked to the underlying orderingof the time-sites.

Now, by using Eq. (62) and considering for simplicity(and ease of notation) a Hilbert H such that

∫dq |q〉〈q| =

1, |ψ〉 = |qi〉, |ϕ〉 = |qf 〉 and Oi(a, a†) ≡ Oi(q) we canwrite

〈qf |T̂O1H(q, t1) . . . OnH(q, tn)|qi〉

=

∫ ∏j 6=0

dqj O1(qj1) . . . O

n(qjn) 〈qfq1 . . . |eiJ �|qiq1 . . . 〉 ,

(67)

where we used the resolution of the identity in H,∫ ∏j dqj |q〉〈q| = 1 (here |q〉 =

⊗j |qj〉tj satisfying (6)).

The right-hand side is formally identical to the standardPI expansion of this quantity for a periodic evolution(such that U†(t, 0) = U(T , t)):

〈qf |T̂O1H(q, t1) . . . OnH(q, tn)|qi〉

=

∫ ∏j 6=0

[ dqj√2πi�m

] 1√2πi�m

O1(qt1) . . . On(qtn) e

iS ,

with S the classical action for H(p, q, t) = p2/2m+V (q, t)(not required in Eq. (67)) evaluated on each path. Re-markably, the quantity eiS is now appearing from thematrix elements of eiJ � along the “quantum trajectories”defined by the extended Hilbert space and representedin Fig. 1. This can be seen explicitly by writing first〈q|eiJ �|q〉 =

∫ ∏j dpj〈q|p〉〈p|eiJ �|q〉, (|p〉 =

⊗j |pj〉tj

satisfying (6)) and noting that (t ≡ j)

〈p|eiJ �|q〉 = ei∑t �[ptq̇t−H(pt,qt,t)]〈p|q〉+O(�2) , (68)

as it follows from the approximation of the operator

eiJ � =1 + i�J +O(�2)

=1 + i�∑t

[ptq̇t −H(pt, qt, t)] +O(�2) ,

where q̇t = (qt+1 − qt)/� + O(�2), i.e. q̇t is equal tothe site derivative of qt in this order (see also Eqs. (A3,A6)). We can corroborate the result (68) by noting thateiJ � = V†eiPt�V = eiPt�

⊗j U(tj + �, tj) which, by con-

sidering again Eq. (66) (now “from right to left”), im-plies 〈p|eiJ �|q〉 =

∏j〈pj |U(tj + �, tj)|qj−1〉, the expres-

sion which is obtained through the conventional time-slicing for a spacing �, in agreement with (68).

Related expressions can be derived for propagators bysimilar means. Coherent states (2) may also be employedanalogously. Since the “sum over trajectories” interpre-tation [55] acquire in H Hilbert space meaning, the con-ventional subtleties of PIs concerning the limits � → 0,T → ∞ appear now linked to standard issues related totensor products in Hilbert space. The results and regu-larizations employed in this work may thus constitute afirst step towards tackling those subtleties by means ofwell-known techniques from canonical QM.

V. DISCUSSION

The treatment presented in this paper provides a start-ing point for developing general space-time formulations.While it is able to reproduce the conventional predic-tions of QM concerning time evolution, it maps the evo-lution to history states endowed with a rich time struc-ture. This natural consequence of the underlying prod-uct structure in time of the extended Hilbert space openssome immediate possibilities concerning the understand-ing of time correlations. In particular, such time struc-ture could be relevant in the investigation of the entan-glement/geometry connection [56–58] since it may en-able space-time extensions of recent proposals of emerg-ing space from entanglement in Hilbert space [59]. Moregenerally, quantum correlations across time-like (causallyconnected) intervals acquire meaning.

In this work, almost all the efforts concerning physicalpredictions have been focused on the recovering of theconventional consequences of QM. However, the unitaryequivalence between theories revealed by the formalismopens additional unexplored possibilities. For example,since all theories are defined in the same Hilbert space H,not only the time evolution of all possible theories followbut, in principle, also that of any quantum superpositionof them (being H a genuine Hilbert space), a situationwhich may find its place in nature: A coherent superposi-tion of gravity [60–62] could induce coherences in the timeevolution of matter. A related example is the possibil-ity of introducing indefinite causal-order (superpositionof causal relations between events), a problem which alsorequires a non-trivial extension of QM, recently underconsideration in the context of process matrices [63]. Inthese new scenarios non-diagonal in time properties of Ptand J , some of which have been discussed in AppendicesB and D, may become relevant.

While the space-time quantum actions have a naturalform for infinite dimensional H (they resemble a classi-cal action), the formalism is completely suitable for finitedimensional systems. For instance, since the general evo-lution of a qubit can be encoded in the first two levelsof an harmonic oscillator, “space-time descriptions” of aqubit can be derived from subspaces of the present H(and apparent generalizations to higher dimensions).

The present formulation also provides a consistentframework for discretizing time. In Sec. IV B this dis-

11

cretization has been related to the conventional time-slicing employed in path integrals through the matrixelements of J . Further developments along this line areunder investigation. We also mention that in case a fun-damental spacing � exists, it would have non-trivial phys-ical implications, as recently shown in [15] through the re-lated quantization techniques considered in section III D.Other insights derived from the formulations consideredthere may be further developed in their “second quan-tized version” as particular instances of this framework.For example, the considerations on Lorentz covariance atthe Hilbert space level described in the PaW extensions[6, 7] for relativistic particles were here generalized to(free) fields.

While we have employed pure states, the mixed casefollows straightforwardly by usual means. Consideringin addition that the treatment of composite systems isimplicit in (1), the formalism should describe measure-ments properly by incorporating the processes involved[1], a strategy recently employed in related constructions[3, 20].

Decoherence functionals can also be derived straight-forwardly from the formalism opening possible connec-tions with Isham’s approach [8] (and modern relatedschemes, e.g. [10, 13]). In particular, it is interestingthat the concept of physical states, which appear nat-urally in the second quantization of parameterized par-ticles (Sec. III D), can be related to such quantity, pro-viding a possible unifying bridge between these differentgeneralizations of QM.

We note finally that while a bosonic formulation wasemployed, the formalism is also suited for fermions:

Given a set of fermions bi such that [bi, b†j ]+ = δij , the

corresponding operators on each slice Bi(t), B†j (t) can

be defined as [Bi(t), B†j (t′)]+ = δ(t − t′)δij , which in

particular implies a Pauli’s exclusion principle in time.Then, main basic results, starting from the unitary rela-tion (12a) between Pt and J (see Appendix B), hold ifwe replace A(t)→ B(t).

In summary, we presented a formulation of QM whichtreats time and “space” on the same footing at theHilbert space level. The concept of time evolution is re-placed by the notion of physical subspaces determined bynew central actors: The space-time quantum actions. Allfamiliar tools of QM can now be applied to this extendedframework, paving the way for a novel understanding ofquantum correlations across time.

ACKNOWLEDGMENTS

We acknowledge support from CONICET (N.L.D.,J.M.M.) and CIC (R.R.) of Argentina. Thiswork was supported by CONICET PIP Grant No.11220150100732.

Appendix A: Regularizations and Notation

In this Appendix we summarize the notation conven-tions we adopt in relation to the regularizations appliedto the extent of time T and the spacing between sites�. For completeness and clarity, explicit expressions andlimits are provided as well.

For finite T ,

A(ωk) =1√T

∫ T/2−T/2

dtA(t)eiωkt (A1)

with ωk =2πkT , such that

A(t) = 1√T

∑k

A(ωk) e−iωkt (A2)

with k ∈ Z and [Ai(ωk), A†j(ωk′)] = δijδkk′ . By Eq. (A2)the “site” derivative

Ȧ(t) := limδt→0

A(t+ δt)−A(t)δt

(A3)

becomes identical with −i√T

∑k ωkA(ωk) e

−iωkt.

In continuous notation, we can rewrite (A2) as A(t) =∫dω√2πA(ω)e−iωt, where A(ω) =

√T2πA(ωk) and

∫dω

stands for 2πT∑k, such that iȦ(t) =

∫dω√2πωA(ω)e−iωt.

For T → ∞ this representation becomes exact, with[Ai(ω), A

†j(ω′)] −→

T→∞δijδ(ω − ω′).

On the other hand, for discrete time (finite �), A(ωk)becomes the discrete FT

A(ωk) =√

�T

∑j

Atjeiωktj , (A4)

where Atj =√�A(tj), tj = �j, k, j = −m, . . . ,m and

T/� = 2m + 1, with [Atj , Atj′ ] = δjj′ , such that Atj =√�T

∑kA(ωk)e

−iωktj . The last expression can also be

used for a continuous t, in which case [Ai(t), A†j(t′)] =

δij1T

sin[π(t−t′)/�]sin[π(t−t′)/T ] −→�→0+

δijδ(t− t′).We can also define the one-body unitary operatorF(tj) = exp[−iA†(ωk′)Mkk

′

tj A(ωk)] with (eiMtj )kk′ =√

�T e−i2π(k+j)k′�/T , such that

F†(tj)A(ωk)F(tj) := At(j+k) (A5)

with F†(tj) = eiPttjF†(0) and F(0) the FT. For finite �,Pt is still defined as Pt =

∑k

2πkT A

†(ωk)A(ωk) where thesum now involves T/� values and

eiPt�Atje−iPt� = Atj+1 . (A6)

Similarly, for a non-trivial theory, the physical foliationoperators used in (51) are defined by

F̃†(tj) = V†F†(tj)V = eiJ tj F̃†(0) . (A7)

12

Appendix B: Unitary Relation between Pt and Jand Additional Properties of Pt

Here some additional properties of the generator oftime translation are presented, starting with the rela-tion between its commutator and the “partial” deriva-tive in time. Immediate (but non trivial) consequencesfollow. Before proceeding, an elementary proof of the re-sult (12a) which only employs Eq. (9) is provided below.

Proof of Eq. (12a). For finite T we assume

T̂ ′ exp[−i∫dt

∫ t0+Tt0

dt′H(A(t),A†(t), t′)] = 1 , (B1)

i.e. U(t0 + T , t0) = 1. Then,

eiPtδtV†e−iPtδt = T̂ ′exp[−i∫dt

∫ t−δtt0

dt′H(A(t),A†(t), t′)]

which holds for T → ∞ when H is well-behaved in thelimit of large times. For δt� 1, it leads to

eiPtδtV†e−iPtδt = eiδt∫dtH(A(t),A†(t),t)V† ,

where we used∫ t−δtt0

dt′H(A(t),A†(t), t′) ≈∫ tt0dt′H(A(t),A†(t), t′) − δtH(A(t),A†(t), t) and

the temporal ordering (the second term is always at timet > t′). In conclusion,

V†e−iPtδtV = e−i[Pt−∫dtH(A(t),A†(t),t)]δt (B2)

implying V†PtV = J .

Notice that this proof only employs properties of V†under time translations. In particular, this means thatit also holds for fermionic systems.

We can actually prove, for a general periodic operator

U = exp[∫dtM(A(t),A†(t), t)] , (B3)

the more general result

UPtU−1 = Pt − i(∂U∂t

)U−1 (B4)

= Pt − i∫dtR(A(t),A†(t), t) , (B5)

which is equivalent to

[Pt,U ] = i∂U∂t

, (B6)

where its partial derivative is defined as

∂U∂t

: = limδt→0

e∫dtM(A(t),A†(t),t+δt) − U

δt

=

(∫dtR(A(t),A†(t), t)

)U (B7)

and R(A(t),A†(t), t) is the operator defined by∂∂t′ e

M(A(t),A†(t),t′) = R(A(t),A†(t), t′)eM(A(t),A†(t),t′).

Proof. Using previous definitions we obtain, up to O(δt),

eiPtδtUe−iPtδt = e∫dtM(A(t+δt),A†(t+δt),t)

= e∫dtM(A(t),A†(t),t−δt)

=

(1− δt

∫dtR(A(t),A†(t), t))

)U

= U − i[U ,Pt]δt (B8)

from which Eqs. (B5)–(B6) directly follow. ForM time-independent, ∂U∂t = 0 and UPtU

−1 = Pt, [Pt,U ] = 0.

Analogously, in agreement with (B7),

[Pt,∫dtM(A(t),A†(t), t)] = i

∫dt∂M∂t

(B9)

with ∂M∂t = limδt→0M(A(t),A†(t),t+δt)−M(A(t),A†(t),t)

δt ,implying Eq. (22) in the limit T →∞.

Since V† is a product in time of operators U(t, t0)we can also write V† = exp[i

∫dtM(A(t),A(t), t)] for∫

dtM(A(t),A(t), t) =∑tM(At,A

†t , t) and U(t, t0) =

eiM(a,a†,t). Then

V†PtV = Pt+i [∫dtM,Pt]+

i2

2![

∫dtM, [

∫dtM,Pt]]+. . . ,

(B10)which is an explicit expansion of (B5). On the otherhand, i ddtU(t) = H(t)U(t), with H(t) = H(a,a

†, t), im-plies

−H(t) =∫ 1

0

exp[isM(t)]M ′(t) exp[−isM(t)]ds

= M ′(t) +i

2![M(t),M ′(t)] +

i2

3![M(t), [M(t),M ′(t)]] + . . .

(B11)

since i ddteiM(t) = −

∫ 10eisM(t)M ′(t)ei(1−s)M(t)ds. By

comparing Eqs. (B10, B11) and considering Eq. (B9),the result V†PtV = Pt −

∫dtH = J is recovered.

This reasoning provides further verification of the re-lated result (18) since we can now write, for ∆t =

t − t0, Ã(t) = ei∫dtMeiPt∆tA(t0)e

−iPt∆te−i∫dtM =

eiJ∆tA(t0)e−iJ∆t with J re-appearing from commuta-

tors between Pt and∫dtM (we used [

∫dtM,A(t0)] =

0, as implied by the initial condition).Furthermore, if we consider more complex operators,

e.g.

U = exp[∫dt1dt2M(A(t1),A†(t1),A(t2),A†(t2), t1, t2)]

a reasoning analogous to Eqs. (B5)–(B7) yields

UPtU−1 = Pt − iR (B12)

with R := i[U ,Pt]U−1 = [( ∂∂t1 +∂∂t2

)U ]U−1. For U her-mitian this defines in general quantum actions

J = Pt − iR (B13)

13

for “exotic” theories non-diagonal in time. It also revealsa great amount of further symmetries of Pt (and hence J )since e.g. in the present case R ≡ 0 for ∂M∂t1 = −

∂M∂t2

asit follows from expanding M near t1, t2. Of course, thiscan be immediately generalized to an arbitrary number oftimes. A basic example of these symmetries is providedby the unitary transformations of the ω = 0 mode. Anon-basic example is provided explicitly in Appendix Dwhere Bogoliubov symmetries are considered.

Appendix C: Normalization in the “ThermodynamicLimit”

Normalization of states for an infinite extent of timeis usually regarded as a subtle aspect of quantum for-malisms of time [20, 64]. In the usual quantum treatmentof reparametrization-invariant systems it also prevents toconsider the physical spaces as proper subspaces leadingultimately to abandoning the role of time as an observ-able. In our proposal these aspects appear in a new formwhich allow a straightforward quantum treatment: Given

e.g. |Ψ〉ω =∑i ψiÃ

†i (ω)|Ω̃〉 and |Φ〉ω′ =

∑i ϕiÃ

†i (ω′)|Ω̃〉,

ω′〈Φ|Ψ〉ω = δ(ω − ω′)〈ϕ|ψ〉 , (C1)

where 〈ϕ|ψ〉 =∑i ϕ∗iψi and the presence of δ(ω − ω′)

(≡ (T/2π)δkk′ for T → ∞) is in accordance with thecontinuum spectrum of J (ω〈Ψ| is an eigenfunctional).Eq. (C1) and obvious generalizations to many parti-cle states, are the continuous ω-equivalent of Eq. (26).The important novelty of the formalism is that not onlyeigenfunctional expansions are well defined but also theirtransformation properties under time translations (sincethe latter are defined in the complete Hilbert space H).This means that if we normalize states by permittingsuperpositions in ω, time evolution is still well defined.Physically, this implies quantum coherences in a quan-tity which in conventional QM is regarded as a “pa-rameter”: e.g., in the case of decoupled oscillators (Eq.(32)), an uncertainty on ω around ω = 0 has the phys-ical meaning of quantum uncertainty in the oscillatorfrequencies ωi0. This also holds in the case of Jrel forδ(ω−ω′)→ δ(m2−m′2) but with an important novelty:The product in the right hand of (C1) is the invariantproduct of scalar QFT [6, 7]. The crucial lesson is thatthe form of the inner product in the physical subspacesmay depend on the choice of J according to its sym-metries and the “parameters” in the Hamiltonian whichacquire quantum coherences.

While the previous considerations allow to explore fea-tures not contemplated in conventional QM, they alsoagree with a more “traditional” approach: If Πp is theprojector in Hp then 〈Φ′|Ψ〉 = 〈ϕ|ψ〉 for |Φ〉 = Πp|Φ′〉which constitutes the generalization of the group aver-aging product [64] to H and its subspaces. Alterna-tively, normal operators can all be equally “smeared”:Ãi(0) → Ã′ =

∫dω′ φ(ω′)Ã(ω′) with

∫dω |φ(ω)|2 = 1,

such that L′(

[∏i(a†i )ni ]|0〉

)=∏i[(Ã

′†i )ni ]|Ω̃〉 implying

〈Φ|Ψ〉 = 〈ϕ|ψ〉 for |Ψ〉 = L′(|ψ〉), |Φ〉 = L′(|ϕ〉).

Appendix D: Linear Symmetries and TimeTranslations for Quadratic J

The diagonal form J =∫dω ω Ψ̃†(ω)Ψ̃(ω) remains

invariant under Bogoliubov transformations(Ã(ω)

Æ(−ω)

)→(U VV ∗ U∗

)(Ã(ω)

Æ(−ω)

), (D1)

which for U, V independent of ω are equivalent toΨ̃(t) →

(U VV ∗U∗

)Ψ̃(t), a linear time independent (in

the normal basis) canonical transformation. This in-

cludes transformations of the form Q̃i(t) → αiQ̃i(t),P̃i(t) → P̃i(t)/αi for αi constant, implying the invari-ance of the “Legendre transform form” (12c). Note also

that Lω =12 [Ψ†(ω)Ψ(ω)−Ψ†(−ω)Ψ̃(−ω)] = a†(ω)a(ω)−

a†(−ω)a(−ω) is an angular momentum-like operator:qxpy − qypx =

a†xay−a†yax

2i = a†+a+ − a

†−a− for (

qµipµ

) =aµ±a†µ√

2and ( ax−iay ) =

a+±a−√2

.

Consider now Eq. (27), i.e. J for quadratic theo-ries. As we have seen, diagonalization can be achievedby linear transformations Ψ̃(t) = W−1(t)Ψ(t) satisfy-

ing iẆ (t) = ΠK(t)W (t). Given the general solutionW (t) = exp(−iΠKt)W0 for a time-independent Hamil-tonian,

eiPt∆tΨ̃(t)e−iPt∆t = exp(−iΠK ′∆t)Ψ̃(t) (D2)

where we used W−1(t)W (t + ∆t)Ψ̃(t) = exp(−iΠK ′∆t)with K ′ = W †0KW0. This is an example of Eq. (58)

and of (D1) with

(U VV ∗ U∗

)= exp(−iΠK ′∆t) and U, V

independent of ω, implying

eiPt∆tΨ̃(ω)e−iPt∆t =

(U 00 U∗

)e−iΠω∆tΨ̃(ω) +(

0 VV ∗ 0

)eiΠω∆tΨ̃(−ω) . (D3)

In particular, for ω = 0, i.e. for physical operators,eiPt∆tΨ̃(0)e−iPt∆t = exp(−iΠK ′∆t)Ψ̃(0), which is Eq.(37). From this result we can also infer the effect oftime translations on physical states by first consideringthe vacuum case. It follows from (D3) that while a time

translation has a non trivial effect on |Ω̃〉, it preserves theseparation between modes with distinct |ω| and in par-ticular between the mode 0 and remaining modes (thisholds for any transformation (D1)):

eiPt∆t|Ω̃〉 = eiH(Ψ̃(0))∆t|0̃〉k=0 ⊗ |Ω′(∆t)〉 , (D4)

for Ã(ω)|Ω̃〉 = 0, Ã(ω = 0)|0̃〉k=0 = 0 and H(Ψ̃(0)) =12Ψ̃†(0)K ′Ψ̃(0). Then, given H(ψ) = 12ψ

†K ′ψ,

〈Φ|eiPt∆t|Ψ〉 = 〈ϕ|eiH(ψ)∆t|ψ〉 × 〈Ω′(0)|Ω′(∆t)〉 (D5)

14

for L(|ψ〉) = |Ψ〉 and L(|ϕ〉) = |Φ〉. This implies Eq.(39).

Appendix E: Time Translations for Time-dependentTheories

Consider the unitary operator

W(∆t) := eiPt∆tV†∆tV (E1)

with

V†∆t := T̂′ exp

[−i∫dt

∫ tt0

dt′H(A(t),A†(t), t′ + ∆t)]

which for H time independent W(∆t) → eiPt∆t. From(12a) it follows that

V†∆tPtV∆t = Pt −∫dtH(A(t),A†(t), t+ ∆t)

= e−iPt∆tJ eiPt∆t ,

implying W†(∆t)JW(∆t) = J and hence

[W(∆t),J ] = 0 . (E2)

We can think that the operatorW(∆t) is translating boththe sites and the time dependence of H.

On the other hand,

VW(∆t)V† = VeiPt∆tV†∆t = V[eiPt∆tV†∆te

−iPt∆t]eiPt∆t

with

eiPt∆tV†∆te−iPt∆t = T̂ ′exp

[i

∫dt

∫ tft

dt′H(A(t),A†(t), t′)],

implying

VW(∆t)V† = eiPt∆t T̂ ′exp[i

∫dt

∫ tft0

dt′H(A(t),A†(t), t′)],

(E3)where tf = t0 + ∆t and we used that the second term,which is equal to

⊗t U†(tf , t0), commutes with Pt.

The result (E3) allows us to write

〈Φ|U(t, t0)|Ψ〉 =

0〈Φ|T̂ ′ exp[− i∑k

∫ tt0

dt′H(A(ωk),A(ωk), t′)]|Ψ〉0

(E4)

for U(t, t0) = F(t0)W†(t − t0)F(t) unitary. This yieldsthe relation

〈Φ|U(t, t0)|Ψ〉〈Ω̃|U(t, t0)|Ω̃〉

=〈φ|U(t, t0)|ψ〉〈0|U(t, t0)|0〉

(E5)

with 〈φ|U(t, t0)|ψ〉 the conventional propagator.

Note also that 〈Ω̃|U(t, t0)|Ω̃〉 = 〈0|U(t, t0)|0〉T/� , whichgeneralizes (F3) since 〈Ω̃|U(t, t0)|Ω̃〉 = 〈Ω̃|W†(t− t0)|Ω̃〉.

With the operator W(∆t) of (E1) we can generalizeEq. (D2) to the time dependent case:

W(∆t)Ψ̃(t)W†(∆t) =eiPt∆tV†∆tΨ(t)V∆te−iPt∆t

=W−1∆t (t)Ψ(t+ ∆t)

with W∆t(t) satisfying iẆ∆t(t) = ΠK(t + ∆t)W∆t(t).

By writing then Ψ(t + ∆t) = W (t + ∆t)Ψ̃(t) and usingW−1∆t (t)W (t+ ∆t) = W (t0 + ∆t, t0) we obtain

W(∆t)Ψ̃(t)W†(∆t) = W (t0 + ∆t, t0)Ψ̃(t+ ∆t) , (E6)

in agreement with conventional time evolution. Sincethe unitary transformation (E6) is a constant Bogoliuvobtransformation, all previous considerations in the ω basishold. This implies the time-dependent versions of Eqs.(37), (38), and (39) for eiPt∆t →W(∆t).

Appendix F: Proof of Eqs. (56)–(57)

In order to prove Eq. (56), we note first that from theresult (12a) (which holds in the form V†eiPt∆tV = eiJ∆tin the discrete case) and [Pt,J ] = 0 it follows, using (15),that

〈Φ(t0)|e−iPt∆t|Ψ(t)〉 = 0〈Φ(t0)|e−iPt∆te−i∑tHt∆t|Ψ(t)〉0

= 0〈Φ(t)|e−i∑tHt∆t|Ψ(t)〉0 (F1)

with Ht ≡ H(At,A†t), ∆t = t − t0 and |Ψ(t)〉 :=F†(t)|Ψ〉, |Ψ(t)〉0 := V|Ψ(t)〉 = F†(t)|Ψ〉0. We now actwith the operators F(t) on the exponential to obtain

〈Φ(t0)|e−iPt∆t|Ψ(t)〉 = 0〈Φ|e−i∑kH(A(ωk),A

†(ωk))∆t|Ψ〉0(F2)

where we are now using a discrete notation for ω (ωk =2πkT ) and we used Eq. (51). The sum involves T/�

terms but only the mode-0 contributes to a non-vacuum

matrix element, i.e. 0〈Φ|e−i∑kH(A(ωk),A

†(ωk))∆t|Ψ〉0 =〈ϕ|e−iH∆t|ψ〉 × [〈0|e−iH∆t|0〉]T/�−1. Since of course thisalso holds for |Ψ〉 = |Φ〉 = |Ω̃〉,

〈Ω̃|e−iPt∆t|Ω̃〉 = [〈0|e−iH∆t|0〉]T/� , (F3)

Eq. (56) is obtained.And to show (57), we write first

〈Φ(t0)|e−iPt∆t|Ψ(t)〉 = 0〈Φ(t0)|Ve−iPt∆tV†|Ψ(t)〉0 =0〈Φ(t)|[eiPt∆tVe−iPt∆t]V†eiPt∆t|Ψ(t0)〉0

where ∆t = t− t0 and where in the last equality we usedF(t0) = e−iPt∆tF(t) and F†(t) = eiPt∆tF†(t0). FromEq. (12a) it follows that eiPt∆tVe−iPt∆t = e−i

∑tHt∆tV

and we finally obtain

〈Φ(t0)|e−iPt∆t|Ψ(t)〉 = 0〈Φ(t)|eiJ∆t|Ψ(t0)〉0 , (F4)

which implies (57).

15

[1] M. A. Nielsen and I. L. Chuang, “Quantum computationand quantum information,” Phys. Today 54, 60 (2001).

[2] W. Pauli and N. Straumann, Die allgemeinen PrinzipienderWellenmechanik (Springer, Berlin, Heidelberg, 1990).

[3] L. Maccone and K. Sacha, “Quantum measurements oftime,” Phys. Rev. Lett 124, 110402 (2020).

[4] P. J. Coles, V. Katariya, S. Lloyd, I. Marvian, andM. M. Wilde, “Entropic energy-time uncertainty rela-tion,” Phys. Rev. Lett 122, 100401 (2019).

[5] F. J. Dyson, “The radiation theories of Tomonaga,Schwinger, and Feynman,” Phys. Rev. 75, 486 (1949).

[6] N. L. Diaz and R. Rossignoli, “History state formalismfor Dirac’s theory,” Phys. Rev. D 99, 045008 (2019).

[7] N. L. Diaz, J. M. Matera, and R. Rossignoli, “Historystate formalism for scalar particles,” Phys. Rev. D 100,125020 (2019).

[8] C. J. Isham, “Quantum logic and the histories approachto quantum theory,” J. Math. Phys. 35, 2157 (1994).

[9] C. J. Isham, N. Linden, K. Savvidou, and S. Schreck-enberg, “Continuous time and consistent histories,” J.Math. Phys. 39, 1818 (1998).

[10] J. F. Fitzsimons, J. A. Jones, and V. Vedral, “Quantumcorrelations which imply causation,” Sci. Rep. 5, 18281(2015).

[11] D. Horsman, C. Heunen, M. F. Pusey, J. Barrett, andR. W. Spekkens, “Can a quantum state over time resem-ble a quantum state at a single time?” Proc. R. Soc. A473, 20170395 (2017).

[12] Z. Zhao, R. Pisarczyk, J. Thompson, M. Gu, V. Vedral,and J. F. Fitzsimons, “Geometry of quantum correlationsin space-time,” Phys. Rev. A 98, 052312 (2018).

[13] Jordan Cotler, Chao-Ming Jian, Xiao-Liang Qi, andFrank Wilczek, “Superdensity operators for spacetimequantum mechanics,” J. High Energy Phys. 2018, 1–57(2018).

[14] I. Kull, P. A. Guérin, and Č. Brukner, “A spacetimearea law bound on quantum correlations,” npj QuantumInf. 5, 1–5 (2019).

[15] G. Wendel, L. Mart́ınez, and M. Bojowald, “Physicalimplications of a fundamental period of time,” Phys. Rev.Lett. 124, 241301 (2020).

[16] L. Chataignier, “Construction of quantum Dirac observ-ables and the emergence of wkb time,” Phys. Rev. D 101,086001 (2020).

[17] R. Gambini, R. A. Porto, J. Pullin, and S. Torterolo,“Conditional probabilities with Dirac observables and theproblem of time in quantum gravity,” Phys. Rev. D 79,041501(R) (2009).

[18] K. V. Kuchař, “Time and interpretations of quantumgravity,” Int. J. Mod. Phys. D 20, 3–86 (2011).

[19] M. Bojowald, P. A. Höhn, and A. Tsobanjan, “Effectiveapproach to the problem of time: general features andexamples,” Phys. Rev. D 83, 125023 (2011).

[20] V. Giovannetti, S. Lloyd, and L. Maccone, “Quantumtime,” Phys. Rev. D 92, 045033 (2015).

[21] A. Boette, R. Rossignoli, N. Gigena, and M Cerezo,“System-time entanglement in a discrete-time model,”Phys. Rev. A 93, 062127 (2016).

[22] A. Boette and R. Rossignoli, “History states of systemsand operators,” Phys. Rev. A 98, 032108 (2018).

[23] A. Nikolova, G. K. Brennen, T. J. Osborne, G. J. Mil-

burn, and T. M. Stace, “Relational time in anyonic sys-tems,” Phys. Rev. A 97, 030101(R) (2018).

[24] L. R. S. Mendes and D. O. Soares-Pinto, “Time as a con-sequence of internal coherence,” Proc. Royal Soc. LondA 475, 20190470 (2019).

[25] A. R.H. Smith and M. Ahmadi, “Quantizing time: Inter-acting clocks and systems,” Quantum 3, 160 (2019).

[26] L. J. Henderson, A. Belenchia, E. Castro-Ruiz, C. Bu-droni, M. Zych, C. Brukner, and R. B. Mann, “Quan-tum temporal superposition: The case of quantum fieldtheory,” Phys. Rev. Lett. 125, 131602 (2020).

[27] A. Valdés-Hernández, C. G. Maglione, A. P. Majtey,and A. R. Plastino, “Emergent dynamics from entangledmixed states,” Phys. Rev. A 102, 052417 (2020).

[28] E. Megidish, A. Halevy, T. Shacham, T. Dvir, L. Dovrat,and H.S. Eisenberg, “Entanglement swapping betweenphotons that have never coexisted,” Phys. Rev. Lett.110, 210403 (2013).

[29] D. Pabón, L. Rebón, S. Bordakevich, N. Gigena,A. Boette, C. Iemmi, R. Rossignoli, and S. Ledesma,“Parallel-in-time optical simulation of history states,”Phys. Rev. A 99, 062333 (2019).

[30] The canonical position eigenstate |q〉 can be expressed as|q〉 =

∑∞n=0〈n|q〉

(a†)n√n!|0〉 = e

−q2/24√π e

− a† 22

+√

2qa† |0〉[31]. The multi-dimensional case |q〉 =⊗

ie−q

2i /2

4√π e−a† 2i2

+√

2qia†i |0〉 ∝ e−

12

[∑i a†i (ai−2

√2qi)|0〉

can be immediately generalized to continuum fields [32].[31] F. Soto-Eguibar and H. M. Moya-Cessa, “Harmonic oscil-

lator position eigenstates via application of an operatoron the vacuum,” Rev. Mex. F́ıs. E 59, 122–127 (2013).

[32] M. Schwartz, Quantum Field Theory and the StandardModel (Cambridge University Press, 2014).

[33] R. B. Griffiths, “Consistent histories and the interpre-tation of quantum mechanics,” J. Stat. Phys 36, 219(1984).

[34] M. Gell-Mann and J. Hartle, “Alternative decohering his-tories in quantum mechanics,” (2019), arXiv:1905.05859[quant-ph].

[35] In Schwinger’s formulation, a complete set of commutingoperators is available on space-like surfaces [36]. This im-plies non-vanishing commutators for causally connectedfield operators. Instead, any unequal-time commutatorbetween A(t) and A†(t) vanishes. Conventional algebrasare recovered “a posteriori” in the physical subspaces.Moreover, the integration in J involves all values of timewithout any reference to particular states.

[36] J. Schwinger, “The theory of quantized fields. I,” Phys.Rev. 82, 914 (1951).

[37] K. Savvidou, “The action operator for continuous-timehistories,” J. Math. Phys. 40, 5657–5674 (1999).

[38] L. C. Céleri, V. Kiosses, and D. R. Terno, “Spin andlocalization of relativistic fermions and uncertainty rela-tions,” Phys. Rev. A 94, 062115 (2016).

[39] I. Bialynicki-Birula and Z. Bialynicka-Birula, “Uncer-tainty relation for photons,” Phys. Rev. Lett 108, 140401(2012).

[40] D. N. Page and W. K. Wootters, “Evolution without evo-lution: Dynamics described by stationary observables,”Phys. Rev. D 27, 2885 (1983).

http://dx.doi.org/10.1063/1.532265http://dx.doi.org/10.1063/1.532265http://dx.doi.org/ 10.1103/PhysRevLett.124.241301http://dx.doi.org/ 10.1103/PhysRevLett.124.241301http://dx.doi.org/10.1103/PhysRevLett.125.131602http://dx.doi.org/ 10.1103/PhysRevA.102.052417http://arxiv.org/abs/1905.05859http://arxiv.org/abs/1905.05859http://dx.doi.org/10.1063/1.533050

16

[41] Y. Aharonov and D. Bohm, “Time in the quantum theoryand the uncertainty relation for time and energy,” Phys.Rev. 122, 1649 (1961).

[42] P. A. M. Dirac, “Generalized hamiltonian dynamics,”Can. J. Math. 2, 129 (1950).

[43] B. S. DeWitt, “Quantum theory of gravity. i. the canon-ical theory,” Phys. Rev. 160, 1113 (1967).

[44] Here δJδÃ(ω)

= ωÆ(ω) = [J , Ã(ω)], an equation whichdefines normal modes.

[45] For qi = (ai + a†i )/√

2, pi = i(a†i − ai)

√2, [qi, pj ] = iδij ,

H = 12

∑i,j tijpipj + vijqiqj + uij(qipj + pjqi) with t, v

symmetric matrices, and the matrices t, v, u straightfor-wardly related to ω0, γ [46].

[46] R. Rossignoli and A.M. Kowalski, “Complex modes inunstable quadratic bosonic forms,” Phys. Rev. A 72,032101 (2005).

[47] We assume now that ΠK(t) has real eigenvalues, which isensured by K(t) positive definite [46], in order to warrantperiodic conditions.

[48] Any time-independent stable quadratic Hamiltonian canbe written in this diagonal normal form by an adequatechoice of operators ai, or equivalently, by a constant BTΨ(t)→W0Ψ(t).

[49] C. Kiefer, “Quantum gravity,” Int. Ser. Monogr. Phys.155, 1–432 (2004).

[50] C. Rovelli and F. Vidotto, Covariant loop quantum grav-ity: an elementary introduction to quantum gravity andspinfoam theory (Cambridge University Press, 2014).

[51] By “second quantization”, we indicate the rigorous math-ematical scheme which, given a certain building block (inthe present case the states |tq〉 and |q〉), allow to constructa Fock space of these indistinguishable elements. See e.g.[65]. Not to be confused with the historical arguments ofQFT [66] .

[52] Vlatko Vedral, “Spacetime as a tightly bound quantumcrystal,” (2020), arXiv:2009.10836 [quant-ph].

[53] N. L. Diaz et al., In preparation.[54] C. Isham, N. Linden, and S. Schreckenberg, “The classi-

fication of decoherence functionals: an analog of gleason’stheorem,” J. Math. Phys. 35, 6360 (1994).

[55] R. P. Feynman, “The principle of least action in quantummechanics,” in Feynman’s Thesis—A New Approach ToQuantum Theory (World Scientific, 2005).

[56] M. Van Raamsdonk, “Building up spacetime with quan-tum entanglement,” Gen. Relativ. Gravit. 42, 2323(2010).

[57] B. Swingle, “Entanglement renormalization and hologra-phy,” Phys. Rev. D 86, 065007 (2012).

[58] T. Nishioka, “Entanglement entropy: holography andrenormalization group,” Rev. Mod. Phys. 90, 035007(2018).

[59] C.J. Cao, S. M. Carroll, and S. Michalakis, “Space fromhilbert space: Recovering geometry from bulk entangle-ment,” Phys. Rev. D 95, 024031 (2017).

[60] C. Marletto and V. Vedral, “Gravitationally induced en-tanglement between two massive particles is sufficient ev-idence of quantum effects in gravity,” Phys. Rev. Lett.119, 240402 (2017).

[61] S. Bose, A. Mazumdar, G.W. Morley, H. Ulbricht,M. Toroš, M. Paternostro, A.A. Geraci, P.F. Barker, M.S.Kim, and G. Milburn, “Spin entanglement witness forquantum gravity,” Phys. Rev. Lett. 119, 240401 (2017).

[62] R. J. Marshman, A. Mazumdar, and S. Bose, “Localityand entanglement in table-top testing of the quantumnature of linearized gravity,” Phys. Rev. A 101, 052110(2020).

[63] E. Castro-Ruiz, F. Giacomini, and Č. Brukner, “Dy-namics of quantum causal structures,” Phys. Rev. X 8,011047 (2018).

[64] J. B. Hartle and D. Marolf, “Comparing formulationsof generalized quantum mechanics for reparametrization-invariant systems,” Phys. Rev. D 56, 6247 (1997).

[65] F. Schwabl, Advanced Quantum Mechanics (Springer,2008).

[66] S. Weinberg, The quantum theory of fields, Vol. 2 (Cam-bridge university press, 1995).

http://dx.doi.org/ 10.4153/CJM-1950-012-1http://arxiv.org/abs/2009.10836http://dx.doi.org/ 10.1103/PhysRevD.95.024031

Space-time Quantum ActionsAbstractI IntroductionII FormalismA A Hilbert Space for Quantum TrajectoriesB Time Translations and Space-time Quantum ActionsC Physical States

III The quadratic caseA Quadratic Space-time Quantum ActionsB Time Structure of Physical StatesC Physical Predictions D Second Quantization of Parameterized Particles and PaW formalism E Relativistic Considerations

IV Recovering physical predictions in the general case A Quantum Foliations1 Propagators2 Observables and Correlation Functions

B Path Integrals from Quantum Trajectories

V Discussion AcknowledgmentsA Regularizations and NotationB Unitary Relation between Pt and J and Additional Properties of Pt C Normalization in the ``Thermodynamic Limit'' D Linear Symmetries and Time Translations for Quadratic JE Time Translations for Time-dependent TheoriesF Proof of Eqs. (56)–(57) References