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Mar
28
comment Covariant Quantisation and the Time Operator in String Theory
Oops, should have read anti-hermitian of course. Thanks @Qmechanic.
Mar
15
comment Covariant Quantisation and the Time Operator in String Theory
Hi, lurscher. You are certainly not referring to the time reversal operator for which your statement is true. However, if the time operator were Hermitean then its eigenvalues and thus its expectation values would be purely imaginary. But we certainly wish to have real numbers for time measurements.
Feb
21
comment Covariant Quantisation and the Time Operator in String Theory
My question was not about how to construct a time operator, but how the fact that there cannot be a hermitian time operator is dealt with in string theory. Also your time operator has limited application but I haven't yet look into it in detail.
Feb
21
comment Covariant Quantisation and the Time Operator in String Theory
QGR, could you be a bit more specific?
Feb
18
comment Covariant Quantisation and the Time Operator in String Theory
Thanks! I should have mentioned the hermiticity of T which I silently assumed. One additional note: $p^0$ does not need to be discrete. It is only the mass spectrum that is discrete. But still $p^0>0$ must hold.
Feb
16
comment Covariant Quantisation and the Time Operator in String Theory
The second part of your answer does not satisfy me. I am aware of the advantages of the covariant quantisation and the use of an reduced product. However, I feel uneasy with this. Using square roots of functionals (which is not defined) as an explanation is not quite the answer I was looking for. Backed up with light-cone quantisation it is clear that a Hilbert space for strings exists. But I do not yet understand in a mathematical rigor way how covariant quantisation works.
Feb
16
comment Covariant Quantisation and the Time Operator in String Theory
Sorry, I didn't know that the impossibility of a time operator in quantum mechanics isn't common knowledge. It dates back to Pauli in 1933 as far as I have figured it out. In a nutshell if there is a commutation relation $[T,H]=i$ this implies that it is possible to generate shifts in the spectrum via $\exp(-i\epsilon T) H \exp(i\epsilon T) = H - \epsilon$. Then, H cannot be bounded from below but instead must be continuous from $-\infty$ to $+\infty$. I still don't see how there can be an operator $P^0$ with a positive spectrum and an operator $X^0$ obeying the above commutation relation.