Textbook quantum mechanics treats time as a classical parameter and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}$, the term $\mathrm{\ensuremath{\Delta}}t$ must be interpreted as a time interval and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the ``time at which an event happens'' by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework we derive here two uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.Textbook quantum mechanics treats time as a classical parameter and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}$, the term $\mathrm{\ensuremath{\Delta}}t$ must be interpreted as a time interval and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the ``time at which an event happens'' by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework we derive here two uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.

arXiv:2108.13974Physical Review ANov 24, 202110 citations

On the Uncertainty Relation between Time and Energy

CRC Press eBooksMay 21, 20047 citations

Energy–time uncertainty relations and time operators

Al. Ivanov

Journal of Mathematical ChemistrySep 22, 20066 citations

1 On the Uncertainty Relation between Time and Energy

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

The physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time-energy relation is much more diﬃcult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schro¨dinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schro¨dinger’s equation.The physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time-energy relation is much more diﬃcult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schro¨dinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schro¨dinger’s equation.

CRC Press eBooksMay 21, 20044 citations

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

Tymiński, Andrzej

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.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.

Dec 2, 20250 citations

Time-Energy Uncertainty Principle in Bilocal Geometry and the Dual Minkowski Structure in

\mathbb{R}^{3,3}

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.