Why isn't there a Time operator in QM?

[pellman]

Hi

As a classic treatise of this problem W. Pauli, in Handbuch der Physik, edited by S. Flugge, Springer, Berlin (1958) Vol.5/1,p.60, showed that a "time operator" T conjugate to a Hamiltonian H could not exist if the spectrum of H is bounded.

Regards.

The mistake here is the identification of the Hamiltonian with the energy. A time operator would be conjugate to

$$E=i\hbar\frac{\partial}{\partial t}$$

, not conjugate to H. The statement of the Schrodinger equation

$$H|\psi\rangle=E|\psi\rangle$$

is not the same thing as saying H = E. It is a restriction on the allowed elements of the Hilbert space. Only those state vectors for which
$$H|\psi\rangle=E|\psi\rangle$$ is true are allowed physical states. If H = E, then $$H|\psi\rangle=E|\psi\rangle$$ is a tautology and would not carry any information.

See http://www.iop.org/EJ/abstract/0305-4470/36/18/317 for more. (Also here http://arxiv.org/pdf/quant-ph/0211047 )

 

[meopemuk]

Eugene, how is this different than saying we can choose an origin, lay out a ruler and read off the positions x and y even if there is no system to observe?

Sure, you can do that. But these readings are absolutely useless. Their knowledge does not tell you anything interesting about physical processes in nature. This is not physics.

On the other hand, measuring positions of real particles (e.g., electrons) does give you a valuable information about their behavior. Recording times of these measurements is also important for understanding electron's dynamics.

Eugene.

 

[meopemuk]

The mistake here is the identification of the Hamiltonian with the energy.

There is no mistake. "Hamiltonian" is just another name for the "operator of energy". The Schroedinger equation

$$H|\psi\rangle= i\hbar\frac{\partial}{\partial t}|\psi\rangle$$

is a mathematical expression of the fact that the Hamiltonian H is the generator of time translations.

Eugene.

 

[dextercioby]

Hi

As a classic treatise of this problem W. Pauli, in Handbuch der Physik, edited by S. Flugge, Springer, Berlin (1958) Vol.5/1,p.60, showed that a "time operator" T conjugate to a Hamiltonian H could not exist if the spectrum of H is bounded.

Regards.

It turns out that Pauli's analysis in not mathematically sound. Actually, the analysis took place in the early 30's when the complex apparatus of Hilbert spaces had not been completed. If there were someone at that time that had the maximal available mathematical knowledge to accomplish Pauli's analysis, then his name could only be Stone, von Neumann who were both MATHEMATICIANS.

Please, see this http://arxiv.org/abs/quant-ph/9908033" published in one of the most prestigious physics journals.

 

[Demystifier]

I hope you agree that in the QM formalism the observed system is represented by a state vector, while the measuring apparatus (or corresponding observable) is represented by a Hermitian operator.

I don't agree. I think that both are described by a state vector. See e.g. von Neumann theory of quantum measurements.

 

[Demystifier]

The mistake here is the identification of the Hamiltonian with the energy. A time operator would be conjugate to

$$E=i\hbar\frac{\partial}{\partial t}$$

, not conjugate to H. The statement of the Schrodinger equation

$$H|\psi\rangle=E|\psi\rangle$$

is not the same thing as saying H = E. It is a restriction on the allowed elements of the Hilbert space. Only those state vectors for which
$$H|\psi\rangle=E|\psi\rangle$$ is true are allowed physical states. If H = E, then $$H|\psi\rangle=E|\psi\rangle$$ is a tautology and would not carry any information.

I fully agree with this.

 

[sweet springs]

Hi.
I try to write the essence of mathematics in discussion by W.Pauli. Let us consider a pair of conjugate Hermite operators A and B, AB-BA=i where hbar=1 for brevity. Let |b> be an eigenstate of B with eigenvalue b, B|b> = b|b>. Making a a unitary transformation U(β)=exp(-iβA) where β is any real number, we get the commutation relation BU(β)-U(β)B=β. Applying it to |b>, BU(β)|b>=(b+β)U(β)|b>. So U(β)|b>=|b+β>, eigenstate of B with eigenvalue b+β. Now we know that B should have continuous spectrum spanning from -∞ to +∞.
Let B a Hamiltonian of the system and A a time operator, Hamiltonian should have continuous spectrum spanning from -∞ to +∞.
Regards.

 

[dextercioby]

Hi.
I try to write the essence of mathematics in discussion by W.Pauli. Let us consider a pair of conjugate Hermite operators A and B, AB-BA=i where hbar=1 for brevity. Let |b> be an eigenstate of B with eigenvalue b, B|b> = b|b>. Making a a unitary transformation U(β)=exp(-iβA) where β is any real number, we get the commutation relation BU(β)-U(β)B=β. Applying it to |b>, BU(β)|b>=(b+β)U(β)|b>. So U(β)|b>=|b+β>, eigenstate of B with eigenvalue b+β. Now we know that B should have continuous spectrum spanning from -∞ to +∞.
Let B a Hamiltonian of the system and A a time operator, Hamiltonian should have continuous spectrum spanning from -∞ to +∞. Regards.

Just this <very essence> is wrong. Your arguments are not mathematically valid. The initial commutation relation takes place in a Hilbert space (actually S), while the rest (underlined), if properly formulated, applies to the functionals' space S' from the Gelfand triple S$$\subset$$H$$\subset$$S'.

Galapon's argument uses no RHS, but ordinary Hilbert spaces.

Pauli's argument is for me the perfect example of <hand waving>. For me <hand waving> = <mathematically incorrect/unsound> = <plain wrong>.

 

[pellman]

I fully agree with this.

Thanks for posting. It is encouraging to know I get it.

 

[sweet springs]

Hi.

Just this <very essence> is wrong. Your arguments are not mathematically valid. The initial commutation relation takes place in a Hilbert space (actually S), while the rest (underlined), if properly formulated, applies to the functionals' space S' from the Gelfand triple S$$\subset$$H$$\subset$$S'.

I feel awkward to learn the Gelfand triple S, it seems to be very deep and difficult.
Please just tell me something about the case that A, B are position and momentum. Ordinary QM texts explain U as the translation operator in coordinate or momentum space. Coordinate and momentum have continuous eigenvalues from -∞　to +∞.　　The Gelfand triple method should be introduced to this common case also?

Regards.

 

[meopemuk]

I don't agree. I think that both are described by a state vector. See e.g. von Neumann theory of quantum measurements.

I don't accept von Neumann's theory of measurement. In my opinion, it is completely unnecessary to invoke dynamical QM formalism to describe the measurement act.

Our argument is philosophical rather than physical, so I don't expect to settle it one way or another. At least we know each other's positions now.

Eugene.

 

[strangerep]

Galapon's argument uses no RHS, but ordinary Hilbert spaces.

I had the same thought when I looked at Galapon's paper.

Pauli's argument is for me the perfect example of <hand waving>. For me <hand waving> = <mathematically incorrect/unsound> = <plain wrong>.

Afaict, Pauli's argument still holds if re-expressed in the more rigorous RHS setting.
The basic problem is that an Hermitian time operator satisfying a CCR with the
Hamiltonian is incompatible with the existence of a lowest-energy state.

 

[strangerep]

I feel awkward to learn the Gelfand triple S, it seems to be very deep and difficult.

It can be -- but the essential idea is not so bad. Try the paper quant-ph/0502053.
There's also a readable introduction in sect 1.4 of Ballentine.

Please just tell me something about the case that A, B are position and momentum. Ordinary QM texts explain U as the translation operator in coordinate or momentum space. Coordinate and momentum have continuous eigenvalues from -∞　to +∞.　　The Gelfand triple method should be introduced to this common case also?

Basically, if you're using the Dirac bra-ket formalism and its associated paraphernalia,
you're already working in rigged Hilbert space (aka Gelfand triple) without realizing it.