# What are the properties of time in quantum mechanics?

1. Oct 8, 2012

### 5ymmetrica1

What are the specific properties of time in quantum mechanics?

One thing I have often wondered is,
Does time exist in multiple quantum states until it is measured? IE. is time an unknown variable until the act of measurement is performed by some kind of measuring device? (a sun-dial, a person, a clock, ect.)

-5ym

2. Oct 9, 2012

### Jano L.

You probably refer to a superposition of basis functions used in quantum theory, used for the total wave function. In case the basis is made of normalized Hamiltonian eigenfunctions (harmonic oscillator), this superposition is written as
$$\psi(\mathbf r,t) = \sum_k c_k(t) \Phi_k(\mathbf r) e^{-iE_k t/\hbar}.$$

In your question you assume that the system can be in multiple quantum states at one time. But this is not a standard understanding of quantum state. In case of the quantum system, we say that its state is described by the function $\psi$; the fact that it can be expressed as a linear combination of other functions does not mean that the system is in multiple states. This is easy to see from the fact that the above sum is not unique. There are many ways to express wave function as a linear combination of basis functions. The choice of the basis is purely conventional.

This is the same as in classical physics. A bell can vibrate in a complicated way described by the same kind of superposition, but it has definite state at any time.

The time of an event in most of physics is a real number dependent on the inertial system chosen, but usually it is not thought to be some object that can be in a "state".

3. Oct 9, 2012

### vanhees71

According to the very foundations of quantum theory, time is (as in classical physics by the way) not an observable but just a parameter ordering events (observations if you wish) according to causality.

4. Oct 9, 2012

### dextercioby

I would argue that not only on a formal/mathematical level, but also from a physical perspective, the introduction of an operator describing time is meaningful, even though not textbook-accepted.

5. Oct 9, 2012

### tom.stoer

do you have a reference with a reasonable approach?

6. Oct 9, 2012

### haushofer

Well, in string theory it is something you definitely do, but in ordinary QM I wouldn't know of any reference.

7. Oct 9, 2012

### tom.stoer

there are a lot of references (google: time operator in quantum mechanics), the question is whether they are reasonable ;-)

8. Oct 9, 2012

### dextercioby

I was only interested in the mathematical aspect of putting such an operator alongside the other ones we normally use.

9. Oct 9, 2012

### tom.stoer

There are reasons why time t cannot be promoted to an observable T with [T,H] = i.

Pauli: "we conclude therefore that the introduction of an operator T must be renounced as a matter of principle, and that time t must necessarily be considered as an
ordinary number."

The main problem is that H is both self adjount and bounded from below whereas T has to be self-adjount but with spectrum equals the real line which can be shown to be incompatible with [T,H] = i. There have been ideas to circumvent these objections, but I am not sure about the current status.

10. Oct 9, 2012

### kith

Would T be a proper observable, if it was bounded from below? If this was the case, one could at least use handwaving arguments like the big bang as the beginning of time.

11. Oct 9, 2012

### dextercioby

Pauli's argument is essentially flawed, since this is non-rigorous (it had been developed before the mathematical theory of unbounded operators in a Hilbert space was created by M.H. Stone and John von Neumann). The mathematical theory has been adressed at length in a thread on Pauli's theorem mentioned in one of my blog items.

As per J. von Neumann's widely accepted reasoning there exists a quantum observable assigned to any linear densly defined (essentially) self-adjoint operator acting on a Hilbert space of pure state representatives.

12. Oct 10, 2012

### vanhees71

Yes, and then if time would be an observable, represented by such an essentially self-adjoint operator by definition it would have to fulfill the canonical commutation relation $[\hat{T},\hat{H}]=\mathrm{i}$ with the hamiltonian, which is also by definition an essentially self-adjoint operator. Then it is clear that both operators would have the entire real axis $\mathbb{R}$ as a spectrum, and that contratdicts the assumption (and observation) that we live in a stable world. That's basically Pauli's argument, and I don't see why it should be wrong, even if he as formulated in the manner physicists do and not within the rigorous language of of functional analysis.

13. Oct 10, 2012

### 5ymmetrica1

$[\hat{T},\hat{H}]=\mathrm{i}$

thanks for all the replies
would you mind explaining this equation in simpler terms if it's possible?, and it's relation to t (time)

I find the discussion incredibly interesting but the math is above my current skill level (just finished highschool).

14. Oct 10, 2012

### tom.stoer

The idea is to introduce T as an operator conjugate to the Hamiltonian H, copying the structure for position x and momentum p.

The momentum operator p generates translations in x, so p is conjugate to x with the well-known commutation relation [x,p] = i.

Of course H generates translations in t, but t is a parameter, not an operator. So one must introduce a new operator T which must be conjugate to H with the same commutation relation, namely [T,H] = i.

The problem is the following:
x (usually) has continuous spectrum equal to the entire real line (there are problems where this is not true!)
p has continuous spectrum equal to the entire real line
T must have continuous spectrum (for t) equal to the entire real line
But H must NOT have a spectrum equal to the entire real line, but the spectrum must be bounded from below

Therefore such an operator T with all the above mentioned properties cannot exist.

@dextercioby: I would be glad to see where exactly Pauli's argument is flawed.

15. Oct 10, 2012

### dextercioby

16. Oct 10, 2012

### tom.stoer

I am perfectöly aware about this discussion, but these proposals for T which try to avoid Pauli's obstruction have slightly different properties:
- point spectrum
- spectrum [0,∞)
- non-hermitean T
- non-invariant commutator subspace

So the statement

remains true.