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Spectrum of a quantum relativistic "distance squared" operator

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This question disusses the same concepts as that question (this time in quantum context). Consider a relativistic system in spacetime dimension $D$. Poincare symmetry yields the conserved charges $M$ (a 2-form associated with Lorentz symmetry) and $P$ (a 1-form associated with translation symmetry). The center-of-mass trajectory $x$ is defined by the equations

$$x \wedge P + s = M$$ $${i_P}s = 0$$

I'm implicitely identifying vectors and 1-forms using the spacetime metric $\eta$

Define $X$ to be the point on the center-of-mass trajectory for which the spacelike interval to the origin is maximal. Substituting $X$ into the 1st equation, applying $i_P$ and using the 2nd equation and $(P, X) = 0$ yields

$$X = -\frac{{i_P}M}{m^2} $$

Passing to quantum mechanics, this equation defines $X$ as vector of self-adjoint operators, provided we use symmetric operator ordering the resolve the operator ordering ambiguity between $P$ and $M$. In particular these operators are defined on the Hilbert space of a quantum mechanical particle of mass $m$ and spin $s$.

What is the spectrum of the operator $X^2$, for given $m$, $s$ and $D$?

This post has been migrated from (A51.SE)
asked Feb 4, 2012 in Theoretical Physics by Squark (1,705 points) [ no revision ]

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