# Time operator, or Time eigenfunctions

• JohnSimpson

#### JohnSimpson

"Time" operator, or "Time" eigenfunctions

We seem to define hermitian operators for momentum, position, energy ect., but we don't really talk about a "Time" operator, or "Time" eigenfunctions. What does time mean in standard quantum mechanics, and why is it different than the above dynamical variables?

Fortunately, it is not necessary. The other operators are sufficiently numerous and are already complicated to deal with.

Bob.

We seem to define hermitian operators for momentum, position, energy ect., but we don't really talk about a "Time" operator, or "Time" eigenfunctions. What does time mean in standard quantum mechanics, and why is it different than the above dynamical variables?

Yes, time is really a special quantity in quantum mechanics and in physics in general.

Momentum, position, energy, etc. are attributes of physical systems. You can say "momentum of the particle is p", "position of the particle is x", etc. meaning that these are measured properties of the given particle. However, you cannot say "time of the particle is t". It is more accurate to say "when particle observable was measured, the laboratory clock showed time t". So, in fact, time is an attribute of the laboratory, observer, or the reference frame. For this reason, time is not an observable, and there is no point to introduce an "Hermitian operator of time". It is quite natural to represent time by a simple numerical parameter, as it is done in both classical and quantum mechanics.

Time features prominently in the definition of both Energy, and momentum operators.

since time is relative for each observer, I think it can only be considered in the context of sequences of events because all observers should agree that event A occurred before event B etc. We do not have a better concept of time currently.

meopemuk is I believe correct for standard quantum mechanics, and this is why quantum mechanics doesn't mesh with relativity. With special relativity, every particle should have it's own time coordinate as it's in 4d spacetime. This produces problems with the standard equations though (such as negative energy states).

As such, general quantum mechanics treats time differently. I believe quantum field theories such as QED have an approach that is more in keeping with special relativity, but not general relativity.

since time is relative for each observer, I think it can only be considered in the context of sequences of events because all observers should agree that event A occurred before event B etc. We do not have a better concept of time currently.

According to relativity, you can't actually make this statement :) Two observers could actually observe 2 events occurring in different orders. Look up the concept of 'simultaneous spaces' for more detail. In this respect, as with all others in 4d spacetime, time is no different from space and the ordering of events changes depending on your current position in spacetime, in the same way that ordering of items in space can change depending on your current position in spacetime.

meopemuk is I believe correct for standard quantum mechanics, and this is why quantum mechanics doesn't mesh with relativity. With special relativity, every particle should have it's own time coordinate as it's in 4d spacetime. This produces problems with the standard equations though (such as negative energy states).

As such, general quantum mechanics treats time differently. I believe quantum field theories such as QED have an approach that is more in keeping with special relativity, but not general relativity.

I disagree that the special role of time violates relativity. True, this special role is inconsistent with the 4D spacetime. However, there is a different (more general) view on the principle of relativity, which does not require the existence of the 4D Minkowski spacetime continuum. This view is based on the supreme role of the Poincare group (Lorentz group + space and time translations) as the group of transformations between different inertial observers. Then fully relativistic quantum mechanics can be formulated in terms of unitary representations of the Poincare group in the Hilbert space. 4D spacetime is not needed for that. This theory was first developed by Wigner in 1939. In 1949 Dirac extended this approach to interacting systems as well. The same approach is applicable to quantum field theories. The best reference is Weinberg's "The quantum theory of fields" vol. 1.

It is true that in QFT quantum fields are formulated as operator functions $$\psi(x)$$on the Minkowski "spacetime". However, there is no reason to identify this abstract 4D continuum with physically observable position and time. In fact, arguments $$x$$ of quantum fields are used as integration variables when operators of observables and the S-matrix are calculated in QFT. So, there is no trace of $$x$$ in final results that can be compared with experiments.

However, there is a different (more general) view on the principle of relativity, which does not require the existence of the 4D Minkowski spacetime continuum. This view is based on the supreme role of the Poincare group (Lorentz group + space and time translations) as the group of transformations between different inertial observers.

I personally think this logic is still problematic, to be used as a design principle.

It basically reduces the observer frame to an arbitrary gauge choice, total lack of physical significance. Only the relations between the choices, manifested by the symmetry transformations that generates all possible choices, are considered physical.

I see no other escape than that the process of actually inferring how different observers or gauges relate, must take place relative to an observer. If it doesn't, then one maintains some realist view of this symmetry.

I guess either one thinks that's fine, or one doesn't. This is fine as an effective framework, but I think this logic aren't going to be very viable when it comes to finding a more coherent framework for QG.

The spacetime realism is removed, but instead we have a symmetry realism, or what I've seen philosophers also call structural realism. I have to agree it's a step in the right direction, but it's far from home.

/Fredrik

Time doesn't really exist in quantum mechanics (also not in classical physics).

Time doesn't really exist in quantum mechanics (also not in classical physics).

There is a standard logic, by which this is usually meant, but there is I think nevertheless I think an open question wether this is an appropriate foundation for a scientific model, because no matter how we argue, various forms of time still fails to go away. I think it's fair to say there are good arguments on both sides.

Here is an interesting talk by Lee Smolin.

On the reality of time and the evolution of laws
"There are a number of arguments in the philosophical, physical and cosmological literatures for the thesis that time is not fundamental to the description of nature. According to this view, time should be only an approximate notion which emerges from a more fundamental, timeless description only in certain limiting approximations. My first task is to review these arguments and explain why they fail..."
-- http://pirsa.org/08100049/

There isn't much doubt that Smolin well understands the standard arguments for that time isn't fundamental, as argued by several people. But he still points out that things are not that easy.

I don't think Smolins talk reflects on all details in the best possible way but it's a start that may provoce some new questions. No offense, but I personally think if someone consider his questions just philosophical baloney, they probably missed the points. It's easy to dismiss his arguments on a first reflection, but the more you think about it, there is something there.

/Fredrik

With special relativity, every particle should have it's own time coordinate as it's in 4d spacetime.

How does one define that time coordinate? For example muons, are particles created in the atmosphere by cosmic rays falling constantly to Earth at around 200,000 miles per second. Since muons move so quickly with respect to the laboratory reference frame, time passes slower in the muon reference frame than in the laboratory rest frame. The muon's internal time coordinate makes sense only in the context of comparing to another reference frame. In and of itself, I don't think one can define such a time coordinate according to relativity at least.

Thanks for pointing out simultaneous spaces, a mind boggling concept!

How does one define that time coordinate? For example muons, are particles created in the atmosphere by cosmic rays falling constantly to Earth at around 200,000 miles per second. Since muons move so quickly with respect to the laboratory reference frame, time passes slower in the muon reference frame than in the laboratory rest frame. The muon's internal time coordinate makes sense only in the context of comparing to another reference frame. In and of itself, I don't think one can define such a time coordinate according to relativity at least.

Thanks for pointing out simultaneous spaces, a mind boggling concept!

Correlated states of entangled particles 'ignore' SR and GR and lose their entanglement simultaneously.

In SR time is a dimension, but to define a relativistic clock, that is to obtain the physical property of time, one need to suppose periodicity. This periodic aspect of the time dimension could be at the origin of the quantum phenomena.

Even for light particle such as the electron, the periodic dynamics (the intrinsic period related to the Compton scale) are to fast to be observed, about 10^-22 sec, and the system appears to have aleatoric behavior. Only resolving time scale smaller that this intrinsic periodicity the time observable make sense, beyond this resolution there is the possibility to describe QM in terms of classical waves.

In SR time is a dimension, but to define a relativistic clock, that is to obtain the physical property of time, one need to suppose periodicity. This periodic aspect of the time dimension could be at the origin of the quantum phenomena.

Do you mean something like a minimum 'clock tick' (periodic-like) is similar to a quantum state change, or something like that? What is your idea?

Not exactly, it is something coming from the old formulation of QM (de Broglie, Bohr, Sommerfeld, Einstein ...). The frequency v of a field gives the energy of the quanta E = h v. But, there is more, imposing periodicity T = 1/ v you actually get the correct energy quantization E_n = n h / T . Changing reference system there is a variation of the time interval T which gives the correct dispersion relations for the energy levels E_n, ecc ... . In this way one finds many many deep analogies with QM.

Not exactly, it is something coming from the old formulation of QM (de Broglie, Bohr, Sommerfeld, Einstein ...). The frequency v of a field gives the energy of the quanta E = h v. But, there is more, imposing periodicity T = 1/ v you actually get the correct energy quantization E_n = n h / T . Changing reference system there is a variation of the time interval T which gives the correct dispersion relations for the energy levels E_n, ecc ... . In this way one finds many many deep analogies with QM.

Correct dispersion levels? Can you expand on that? Are you talking about the Von Neumann / Birkhoff Quantum Logic?
http://en.wikipedia.org/wiki/Quantum_logic

A quantized field has an (ordered) energy spectrum E_n = n h / T, because it is like a quantum harmonic oscillator with periodicity T. But you get the same energy spectrum imposing a periodicity T to a wave. The allowed frequency are v_n = n v where v is the fundamental frequency T= 1/ v. Now, if the field has mass M, according to SR the period T change to T' = T / \sqrt{p^2 + M^2} so the energy levels has the relativistic dispersion relation E_n = h n \sqrt{p^2 + M^2}. This quantization is the one you get from the usual QFT. But you can obtain commutation relation or the Schrodinger eq as well. That is assuming periodicity as a principle you get quantization.

We seem to define hermitian operators for momentum, position, energy ect., but we don't really talk about a "Time" operator, or "Time" eigenfunctions. What does time mean in standard quantum mechanics, and why is it different than the above dynamical variables?

I was taught that time is just an independent parameter in standard QM. This is explicitly stated in "Modern Quantum Mechanics" by J.J. Sakurai.

Time is fundamentally different from dynamic variables in that the later are functions of the former. Although there is no time operator associated with measuring the time-state of a system, there is a time evolution operator which allows the description of how the state of a system changes with time.

I probably did not say that in the strictly correct way, but hopefully it makes some sense.

I was taught that time is just an independent parameter in standard QM. This is explicitly stated in "Modern Quantum Mechanics" by J.J. Sakurai.

Time is fundamentally different from dynamic variables in that the later are functions of the former. Although there is no time operator associated with measuring the time-state of a system, there is a time evolution operator which allows the description of how the state of a system changes with time.

I probably did not say that in the strictly correct way, but hopefully it makes some sense.

In the ordinary (Hamiltonian) formulation time appears as an "independent parameter", but in the covariant (Feynman) formulation of quantum mechanics time is a "dynamical variable". However, also in this formulation the quantum time is not exactly the relativistic time. In the latter case SR is invariant under time inversion. The former case, because of the close analogy of the path integral with a partition function, time appears as statistical time which arrow is fixed by the second principle of thermodynamics. These incongruences seems to be solved assuming time as a microscopically and dynamically compact dimension.

There's a kind of flat-space-quantum-state-correlation time that is simultaneous everywhere. It does not need to hold to SR and GR because no information is being transmitted (no causal problems). How? Take 5 entangled particles anywhere in our spacetime universe. When their state correlations collpase that happens at all 5 points simultaneously regardless of what the clocks say on the respective laboratory benches. They are all at the same moment as though they were not separated at all. IMO its a more fundamental time than einsteins spacetime.

Has anybody noticed my post #8?

I did it was an excellent paper, however it concluded on a sour note, with the time eigenstates not being physical. in reference two of that paper however the author demonstrates pauli's proof that time is not an operator, and then makes his way around the problem. I'm surprised that the paper "quantized time" hasn't been given more notice.

I did it was an excellent paper, however it concluded on a sour note, with the time eigenstates not being physical. in reference two of that paper however the author demonstrates pauli's proof that time is not an operator, and then makes his way around the problem. I'm surprised that the paper "quantized time" hasn't been given more notice.
Yes, quantized time seems reasonable to me. There seems to be a lot of debate about time -
in entanglement there seems to be a simultaneous time (ie a 'present moment' rather than spacetime relative times) that correlated states understand. Is this concept used in QFT?

Look, when we go from Classical Mechanics to Quantum Mechanics, we factually go to a wave mechanics with the energy quantization as an eigenvalue problem (consider waves in a box, for example). There is no need to quantize the time in such a wave mechanics.

Bob.

Look, when we go from Classical Mechanics to Quantum Mechanics, we factually go to a wave mechanics with the energy quantization as an eigenvalue problem (consider waves in a box, for example). There is no need to quantize the time in such a wave mechanics.

Bob.

The same quantization can be obtained by imposing periodic BCs in time, that is the time is in a (temporal)box with length T=h/E being E the energy of the quanta.

Look, when we go from Classical Mechanics to Quantum Mechanics, we factually go to a wave mechanics with the energy quantization as an eigenvalue problem (consider waves in a box, for example). There is no need to quantize the time in such a wave mechanics.
Yes, but relativity suggests that time should be treated on an equal footing with space. So if there is a space-position operator, there should also be a time-position operator.

There's a kind of flat-space-quantum-state-correlation time that is simultaneous everywhere. It does not need to hold to SR and GR because no information is being transmitted (no causal problems). How? Take 5 entangled particles anywhere in our spacetime universe. When their state correlations collpase that happens at all 5 points simultaneously regardless of what the clocks say on the respective laboratory benches. They are all at the same moment as though they were not separated at all. IMO its a more fundamental time than einsteins spacetime.

You've 5 events in M4 labeled as "simultaneous" but they do not occupy a single hypersfc, so what do you mean by "simultaneous?" Proper time for all is the same at these events? That would seem problematic since it suggests a worldline and you can't have worldlines for screened-off entities (right?).

Yes, but relativity suggests that time should be treated on an equal footing with space. So if there is a space-position operator, there should also be a time-position operator.

If there is time operator, then it should have eigenstates. So, there should be, for example, a state in which an electron is "localized" in time at t=0. Then the probability of finding the electron at t<0 or at t>0 is zero. This seems to contradict the charge conservation law.

P764RDS: Quote
"There's a kind of flat-space-quantum-state-correlation time that is simultaneous everywhere. It does not need to hold to SR and GR because no information is being transmitted (no causal problems). How? Take 5 entangled particles anywhere in our spacetime universe. When their state correlations collpase that happens at all 5 points simultaneously regardless of what the clocks say on the respective laboratory benches. They are all at the same moment as though they were not separated at all. IMO its a more fundamental time than einsteins spacetime."

Ruta:
You've 5 events in M4 labeled as "simultaneous" but they do not occupy a single hypersfc, so what do you mean by "simultaneous?" Proper time for all is the same at these events? That would seem problematic since it suggests a worldline and you can't have worldlines for screened-off entities (right?).

No, no worldlines here:
There is no proper time needed for the 5 decorrelation events. Proper time is irrelevant here. Also:
1) Do not need to obey the laws of physics as in einstein's SR. These are quantum state correlations and not laws of physics in the normal sense.
2) The speed of correlation knowledge (not speed of information!) is infinite - not finite as in light/gravitons information maximum speeds. Because of the instantaneous nature there is only one reference frame - its a kind of Newtonian/Galilean type of frame. Not Riemann or Minkowski.
3) The proper time is unknown and not needed.
4) The simultaneous decorrelations of entangled quantum states between separated entangled particles or a 'present moment' is required.
5) The speed of 'correlation knowledge' is infinite in this model (a *better* view is: there is no perceived separation between the correlated states of entangled particles (hence simultaneous) Note: The particles themselves obey normal SR, GR and reference frame physics, its only the correlated states we are concerned with)

"There is no proper time needed for the 5 decorrelation events. Proper time is irrelevant here. Also:
1) Do not need to obey the laws of physics as in einstein's SR. These are quantum state correlations and not laws of physics in the normal sense."

The detection events occur somewhere in spacetime, regardless of QM, so what do you mean by "simultaneous" if not in the context of spacetime?

If there is time operator, then it should have eigenstates. So, there should be, for example, a state in which an electron is "localized" in time at t=0. Then the probability of finding the electron at t<0 or at t>0 is zero. This seems to contradict the charge conservation law.
As explained in the mentioned paper, one should make a difference between kinematics and dynamics. The kinematic time operator exists and it does have eigenstates - delta functions in time. However, such wave functions do not satisfy the standard wave equations of motion, so eigenstates are "unphysical" from the dynamical point of view. If, for example, tachyons existed, then time eigenstates would be dynamically physical, while space eigenstates wouldn't.

But the main point is the following. Kinematics should be defined BEFORE the dynamics. In this sense the time operator exists, because the operators belong to kinematics, not dynamics.

As explained in the mentioned paper, one should make a difference between kinematics and dynamics. The kinematic time operator exists and it does have eigenstates - delta functions in time. However, such wave functions do not satisfy the standard wave equations of motion, so eigenstates are "unphysical" from the dynamical point of view. If, for example, tachyons existed, then time eigenstates would be dynamically physical, while space eigenstates wouldn't.

But the main point is the following. Kinematics should be defined BEFORE the dynamics. In this sense the time operator exists, because the operators belong to kinematics, not dynamics.

Hi Demistyfier,

your paper is based on an (unproved) assumption that time and space "should be treated on an equal footing". In my opinion, this contradict the physical meanings of the notions "time" and "position". Position is a property of a particle (or any other physical system). That's why we call it "observable" together with momentum, spin, energy and other observables. However time is not a property or attribute of the particle/system. In order to "measure time" you just need to look at your watch. This value does not depend on the physical system that you are observing. You even may not have any physical system or particle in your laboratory - and you can still "measure time". So, "time" is not an observable in the usual sense.

Position is a property of a particle (or any other physical system). That's why we call it "observable" together with momentum, spin, energy and other observables. However time is not a property or attribute of the particle/system. In order to "measure time" you just need to look at your watch. This value does not depend on the physical system that you are observing. You even may not have any physical system or particle in your laboratory - and you can still "measure time". So, "time" is not an observable in the usual sense.

I don't follow how this argument proves that time is fundamentally different from space.
Certainly, in the laboratory, we can only note correlations (in Rovelli-esque meaning),
but surely this also applies to position? Consider the original Rutherford scattering
experiments, where he had some assistants sitting in a dark room recording the position
of flashes on a screen (within a system of grid regions on the screen). If one removes the
particle source, the screen is still there and the positions of various grid regions on that
screen still exist as references. In this sense, we can still "measure position" even if there's
no physical experimental system in the lab. I don't see how this is fundamentally from
"measuring time" by looking at one's watch, except perhaps in how the watch's reading
advances continually while we watch it.