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Time-Energy Uncertainty Principle in Bilocal Geometry and the Dual Minkowski Structure in $\mathbb{R}^{3,3}$

Tymiński, Andrzej

Published Dec 2, 2025• 0 citations
Publisher

Abstract

In the bilocal space $\mathbb{R}^{3,3}$, quantum time can be defined as the distance between two states of the same particle. This construction leads to the Minkowski interval$ds^2 = t^2 - r^2$and to the interpretation of $t$ as a time coordinate emerging geometrically. In the present paper we develop the dual side of this construction by introducing the gradient energy operator$\hat{E} = -i\hbar(\nabla_2 - \nabla_1).$We show that the time and gradient energy operators form a canonically conjugate pair:$[\hat{t},\hat{E}] = 2 i \hbar,$which leads to a relational uncertainty principle of the time--energy type$\Delta t \,\Delta E \gtrsim \hbar.$At the same time the intervals $(t,r)$ and $(E,p)$ form two equivalent Minkowski spacetimes arising from a single bilocal geometry. This formalism reveals a hidden geometric structure of quantum theory, in which time, energy and mass appear as emergent quantities resulting from relations between states.

DOI
10.5281/zenodo.17784113

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Time-Energy Uncertainty Principle in Bilocal Geometry and the Dual Minkowski Structure in R3,3\mathbb{R}^{3,3}R3,3
In the bilocal space R3,3\mathbb{R}^{3,3}R3,3, quantum time can be defined as the distance between two states of the same particle. This construction leads to the Minkowski intervalds2=t2−r2ds^2 = t^2 - r^2ds2=t2−r2and to the interpretation of ttt as a time coordinate emerging geometrically. In the present paper we develop the dual side of this construction by introducing the gradient energy operatorE^=−iℏ(∇2−∇1).\hat{E} = -i\hbar(\nabla_2 - \nabla_1).E^=−iℏ(∇2​−∇1​).We show that the time and gradient energy operators form a canonically conjugate pair:[t^,E^]=2iℏ,[\hat{t},\hat{E}] = 2 i \hbar,[t^,E^]=2iℏ,which leads to a relational uncertainty principle of the time--energy typeΔt ΔE≳ℏ.\Delta t \,\Delta E \gtrsim \hbar.ΔtΔE≳ℏ.At the same time the intervals (t,r)(t,r)(t,r) and (E,p)(E,p)(E,p) form two equivalent Minkowski spacetimes arising from a single bilocal geometry. This formalism reveals a hidden geometric structure of quantum theory, in which time, energy and mass appear as emergent quantities resulting from relations between states.
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It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.
A derivation from first principles is given of the energy-time uncertainty relation in quantum mechanics. A canonical transformation is made in classical mechanics to a new canonical momentum, which is energy E, and a new canonical coordinate T, which is called tempus, conjugate to the energy. Tempus T, the canonical coordinate conjugate to the energy, is conceptually different from the time t in which the system evolves. The Poisson bracket is a canonical invariant, so that energy and tempus satisfy the same Poisson bracket as do p and q. When the system is quantized, we find the energy-time uncertainty relation \ensuremath{\Delta}E\ensuremath{\Delta}T\ensuremath{\ge}\ensuremath{\Elzxh}/2. For a conservative system the average of the tempus operator T^ is the time t plus a constant. For a free particle and a particle acted on by a constant force, the tempus operators are constructed explicitly, and the energy-time uncertainty relation is explicitly verified.
ABSTRACT The emergence of quantum entanglement correlations across spatially separated measurement regions renders the interpretation of the relationship between quantum mechanics and relativistic causality a core problem in quantum foundations [Einstein et al., 1935; Bell, 1964]. In this study, two entangled particles are modeled not as two independent physical objects in Minkowski spacetime, but as a joint quantum state in configuration space. The relative space and time coordinates of the joint state are defined via: and the ideal space-time correlation limit assumed in the model is expressed as: This limit is not a general definition of quantum entanglement, but an idealized geometric assumption representing strong correlation in the relative coordinates of the joint state within the proposed model. In this approach, physical spacetime is treated as a geometric structure representing the projection of the joint quantum state onto observable spacetime coordinates. To ensure the projection is Lorentz-covariant, the light-like interval: is employed as the constitutive geometric condition of the model. This condition is accepted not as a mandatory Minkowski interval between physical measurement events, but as the mathematical postulate of the proposed projection geometry. The dynamic scale parameter of the model is grounded in total relativistic quantum energy instead of rest mass, establishing a dimensionless Planck-scaling factor defined as: To encompass both massive and massless systems, quantum energy density is defined via the energy density component of the stress-energy tensor relative to an observer. The required consistency condition for covariant energy-momentum conservation is derived in its general form. Furthermore, modeling the proposed geometric effect as a dimensionless metric perturbation yields a phenomenological first-order geometric phase correction defined as: where is the energy-dependent reduced wavelength. Under fixed and finite , the vanishing of the proposed correction in the low-energy regime relative to the Planck scale () demonstrates that the geometric effect added by the model to standard low-energy quantum mechanics becomes negligible. This study proposes a testable theoretical model aimed at explaining the spatiotemporal correlations of quantum entanglement within a Lorentz-covariant, energy-scaled, and phenomenological geometric framework.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.