A straightforward extension of quantum mechanics and quantum field theory is proposed that can describe physical systems comprising two interacting subsystems: one subsystem evolves forward in time, the other, backward. The space of quantum states is the direct sum of the states representing the respective subsystems, whereupon there are two linearly independent vacuum states, one each for the forward and the backward subspace. An indefinite metric is imposed on the space of quantum states such that purely forward (respectively, purely backward) states have positive (respectively, negative) norms. Hamiltonians are self‐adjoint operators with respect to the metric, such that interactions/transitions between the subspaces can be accounted for. Given suitable definitions of input and of output states at the two ends of a time interval, input and output states separately have positive norms such that probability is conserved, and hence S‐matrices are unitary. A discussion of the physics entailed in the proposed formalism is undertaken. Then as an application, a simple model of a relativistic quantum field theory is proposed. In this model, the expected vacuum energy (thought to be associated with the cosmological constant) almost vanishes uniformly for times in an interval due to cancellation of the energies of the forward and backward vacuum states; this cancellation holds whatever be the input vacuum state at the ends of the time interval. A model is advanced wherein magnetic monopoles live exclusively in backward‐evolving states, and interact with forward‐evolving electric charges in a certain way. Proposals for further research, particularly concerning the possible detection of advanced gravitational waves, and a conjecture on the physics of dark matter and dark energy, conclude the report.
A straightforward extension of quantum mechanics and quantum field theory is proposed that can describe physical systems comprising two interacting subsystems: one subsystem evolves forward in time, the other, backward. The space of quantum states is the direct sum of the states representing the respective subsystems, whereupon there are two linearly independent vacuum states, one each for the forward and the backward subspace. An indefinite metric is imposed on the space of quantum states such that purely forward (respectively, purely backward) states have positive (respectively, negative) norms. Hamiltonians are self‐adjoint operators with respect to the metric, such that interactions/transitions between the subspaces can be accounted for. Given suitable definitions of input and of output states at the two ends of a time interval, input and output states separately have positive norms such that probability is conserved, and hence S‐matrices are unitary. A discussion of the physics entailed in the proposed formalism is undertaken. Then as an application, a simple model of a relativistic quantum field theory is proposed. In this model, the expected vacuum energy (thought to be associated with the cosmological constant) almost vanishes uniformly for times in an interval due to cancellation of the energies of the forward and backward vacuum states; this cancellation holds whatever be the input vacuum state at the ends of the time interval. A model is advanced wherein magnetic monopoles live exclusively in backward‐evolving states, and interact with forward‐evolving electric charges in a certain way. Proposals for further research, particularly concerning the possible detection of advanced gravitational waves, and a conjecture on the physics of dark matter and dark energy, conclude the report.
Schroumldinger equation for joint bidirectional motion in time
The conventional time-dependent Schrödinger equation describes only unidirectional time evolution of the state of a physical system, i.e. forward or, less commonly, backward. This paper proposes a generalized quantum dynamics for the description of joint, and interactive, forward and backward time evolution within a physical system. The principal mathematical assumption for bidirectional evolution in general is that the space of states should be taken to be not merely a Hilbert space, but a more restricted entity known as a Kreĭn space, which is a complex Hilbert space with a Hermitian operator that has eigenvalues +1 and −1 only, and that therefore gives rise to an indefinite metric. The vector subspaces of states with positive or negative norm with respect to the indefinite metric will—for open channels—be construed to be states in forward or, respectively, backward evolution along the time axis. The quantum dynamics is generated by a pseudo-Hermitian Hamiltonian operator and conserves inner products with respect to the indefinite metric. Input and output states are defined in physically plausible ways such that the output comprises both reflected and transmitted states from a zone of interaction in time; a unitary transformation between input and output states is obtained from the pseudounitary transformation between the initial and final states. Three applications are studied: (1) a formal theory of collisions in terms of perturbation theory; (2) a relativistically invariant quantum field theory for a system that kinematically comprises the direct sum of two quantized real scalar fields, such that one subfield evolves forward and the other backward in time, and such that there is dynamical coupling between the subfields; (3) an argument that in the latter field theory, the dynamics predicts that in a range of values of the coupling constants, the expectation value of the vacuum energy of the universe is forced to be zero to high accuracy.
Time-symmetric relativity theory (TSR) based on Zisman-Stueckelberg-Feynman (ZSF) interpretation includes reference frames going backward in time, maximally expanding their symmetry group to general Lorentz group O(1,3) with 4-inversion (Article 1). Similar time-symmetric (TS) extensions of relativistic quantum mechanics (TS RQM) (Article 2) and QFT (TS QFT) (this article) avoid problems of former RQM and QFT. Former QFT introduced unfounded hypotheses about physical quantum fields in entire space for each type of particle, and these hypotheses led to insoluble problems. TS QFT is limited to known external interaction fields, and describes particles using theory of many-particle systems. In second quantization method, single-particle wave functions and operators, which increase or decrease number of particles in system, form operator amplitudes of transition probabilities. These amplitudes are complex, act in their state spaces for given energy sign, and lead to positive state norm. As in quantum mechanics, they, together with conjugate amplitude, form canonical pair. Canonical formalism in relativity theory is consistent at using Hamilton-Jacobi (HJ) formalism, where 3-momentum and Hamiltonian enter into HJ equation in the same degree. Differential form of relativistic HJ equation is Klein-Gordon (KG) equation, expressing the relationship between energy and momentum, and is general equation for physical components of all operator amplitudes. For bosons, KG equation suffices; for fermions, linearization of HJ equation is required, leading to special first-order equations, such as Dirac equation. Operators of observables are normally ordered, and there are no zero-point fluctuations in ground states. Probability current is determined by 4-momentum and has the same sign; interaction currents are also proportional to probability current and change sign with 4-momentum, which corrects the currents in fermion theory. Introducing interactions into HJ equation leads to new contact terms. Time-symmetric orderings of operator amplitudes lead to causal propagators and usual diagram technique. Taking into account external gravitational field of quanta at Planck length, gravitational radius of the system, and dilation of their proper times in this field lead to gravitational self-regularization of loop diagrams. This makes perturbation theory series in Standard Model (SM) and quantum gravity finite and convergent. Contributions of chiral anomalies are finite and cancel out in SM.
Quantum mechanics is traditionally formulated using the reversible coordinate time t, which enforces a globally reversible unitary group structure and time‑symmetric propagators. This conflicts with physically irreversible processes such as decoherence, measurement, entropy production, causal propagation, and curvature‑driven focusing in general relativity. We resolve this mismatch by replacing t with the monotonic temporal field T, previously developed in the context of global causal structure, temporal focusing, and worldline ordering in curved spacetime. Once T is taken as the fundamental evolution parameter, quantum dynamics cannot be reversible: the inverse operator U−1 is not physically realizable, and unitary evolution becomes a forward‑directed semigroup. We formulate **forward unitarity**, in which physical evolution is generated only by forward‑directed operators Uphys(T2,T1) with T2>T1. Pure states evolve according to a forward Schrödinger equation with an intrinsic, monotonic decoherence functional D(T), while mixed states evolve under a forward Lindblad structure. Probability is conserved, but coherent recoverability is forbidden by geometric irreversibility, causal structure, and temporal focusing. We construct a Forward‑QM Lagrangian, a forward Heisenberg picture, and a forward path integral restricted to histories with non‑decreasing T, showing how operator algebras, propagators, correlation functions, and creation/annihilation operators inherit the arrow of time. Forward‑QM provides a unified, forward‑directed formulation of quantum mechanics compatible with general relativity, causal geometry, and the irreversible structure of spacetime. It replaces reversible unitary groups with physically meaningful semigroup evolution, aligns quantum dynamics with measurement, decoherence, and entropy production, and establishes the foundation for Forward Quantum Field Theory (FQFT). The extension to FQFT—covering forward propagators, interacting fields, renormalization in the presence of monotonic decoherence, and causal compatibility in curved spacetime—will be developed in future work.
We introduce an embedding of the Klein-Gordon equation into a pair of coupled equations that are first-order in time. The existence of such an embedding is based on a positivity property exhibited by the Klein-Gordon equation. These coupled equations provide a more satisfactory reduction of the Klein-Gordon equation to first-order differential equations in time than the Schrodinger equation. Using this embedding, we show that the ``negative probabilities" associated with the Klein-Gordon equation do not need to be resolved by introducing matrices as Dirac did with his eponymous equation. For the case of the massive Klein-Gordon equation, the coupled equations are equivalent to a forward Schrodinger equation in time and a backward Schrodinger equation in time, respectively, corresponding to a particle and its antiparticle. We show that there are two positive integrals that are conserved (constant in time) in the Klein-Gordon equation and thus provide a concrete resolution of the historical puzzle regarding the previously supposed lack of a probabilistic interpretation for the field governed by the Klein-Gordon equation. A significant consequence is that the Schrodinger equation is given a relativistic formulation, which does not require creation and annihilation operators, i.e. quantum fields. Physically, this corresponds to a theory in which the positive and negative energy parts do not directly interact, hence there will be no annihilation events--for example, particle-antiparticle collisions which do not result in photon emission. Thus, one practical consequence of this relativistically consistent theory is a simple explanation for dark matter.
In this chapter, we finally introduce time evolution. The development is carried out for finite, enumerable systems (i.e., systems whose states live in a finite dimensional, inner product space) which, of course, allows a straight forward extension to the continuum and infinite dimensional state spaces. The necessary mathematical background of unitary operators, and their properties relevant to quantum mechanics are first introduced. In particular, the fact that preservation of norm under the action of an operator implies unitarity, and that every unitary operator can be expressed as the exponential of i timesa Hermitian operator are introduced. Time evolution is then introduced in its integral form through the time evolution operator, and the Hamiltonian operator is introduced in this context. This approach, however, necessitates the restriction to conservative systems. Next, the differential form of the law of time evolution, i.e., Schrodinger equation, is derived from the time evolution operator, and this is postulated to be the general law of time evolution. We then demonstrate how, for conservative systems, the solution of the Schrodinger equation essentially reduces to solving the eigenvalue problem of the Hamiltonian. Constants of motion and Heisenberg picture have been relegated to the problem section.
Time-symmetric theory (TST) is formulated as a theoretical foundation of time-symmetric physics (TSP), classical, relativistic and quantum. TST is based on the widely used in particle physics fact that the description of positive energy antiparticles moving forward in space and time is equivalent to the description of negative energy particles moving backward in space and time (Zisman 1940, Stückelberg 1941, Feynman 1949). In TST, this fundamental fact is generalized by formulating the equivalence principle of particle physics and applying it to all physical phenomena in the rest frames of negative energy particles. As a result, the group of transformations TST also includes such inertial frames that move backward in space and time with coordinates related to the ordinary ones by 4-inversion. This led to the formulation of time-symmetric relativity theory and corresponding relativistic quantum mechanics. In the latter, the probabilities remain positive, but the bilinear forms of the wave functions are associated with probability flows, the signs of which depend on the direction of movement along the coordinate axes, including the time axis. In TST, therefore, the Klein-Gordon equation is a consistent equation for probability amplitudes, and the fermion theory contains the necessary corrections, which make it physically more consistent, unchanging its main observational consequences. Applications of TST to quantum fields will be considered in the second article.