no time operator

2007-05-27

According to a theorem established by W. Pauli in the late 1920s, there cannot be a
time-operator in quantum theory. It is fairly easy to see why this is so:
Assume that an operator T exists, with [T,H] = iI. Consider then the unitary operator
U = exp(-ibT), with b an arbitrary real number. The one gets [U,H] = -bU, using a
straightforward expansion of the exponent.
Let |Psi> be an eigenstate with energy E, then HU|Psi> = (E+b)U|Psi>, so that U|Psi>
is a state with energy E+b, where b was an arbitrary real number.
In other words, T can only exist if the energy spectrum is continuous and in particular
not bounded below. Pauli concluded that
"the introduction of an operator T must fundamentally be abandoned and that the time
 t in quantum mechanics has to be regarded as an ordinary number."

Since then, time in quantum theory is what one reads from a classical clock.
But like all no-go theorems, this argument might have loopholes which one could try to
explore. Perhaps one can make sense of a quantum theory with unbounded energy?
After all, Dirac's electron theory has an unbounded energy spectrum, but with the added
assumption that in the real vacuum all states with E < 0 are already occupied; The famous
'hole theory' which predicted the existence of positrons.
The paper quant-ph/0702111 examines another proposal to work around Pauli's theorem;
Even more about this is discussed in quant-ph/9908033.


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