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S2i. Position operators in relativistic quantum field theory
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In relativistic quantum field theory in its usually given form,
position is promoted to the same status as time, and hence becomes a
parameter in the qwuantum field, while in quantum mechanics it is an
operator vector.
This poses the question of whether there is a position operator in
relativistic quantum field theory. Many people think that there is none.
But even though there is a parameter called x and referred to as
4-dimensional position, there is also an vector defining a
3-dimensional position operator, provided the relativistic system
under consideration is not massless.
Indeed, any relativistic theory possesses the Poincare group as a
symmetry group, whose infinitesimal generators satisfy the standard
commutation rules of the Poincare algebra. But given these, the
standard construction by Newton and Wigner gives (in each Lorentz
frame) a 3-dimensional position operator with commuting components,
and the associated conjugate momentum operators. (See Section S2g
''Particle positions and the position operator'' of this FAQ.)
These play exactly the same role as the position and momentum
operators in nonrelativistic quantum mechanics.