An exceptional quantum mechanics in 26 dimensions?
One of the axioms of field theory is that the amplitude for a path is
where L is distance from x to y.
L is an invariant of O(n) where n is the number of dimensions.
One might ask, it possible to have quantum mechanics in space where the symmetry is a different group? Sp(n) and SU(n) have invariants of the form xy-yx, which unfortunately are identically zero for commuting coordinates. So really we are only interested in groups that can be defined in terms of symmetric invariants.
An interesting group is F4. In 26 dimensions it has the symmetric invariant (S):
So we could define a quantum theory in this space with an invariant based on these two invariants. For example we could define our amplitude as:
exp(i L )
where L is distance from x to y.
L is an invariant of O(n) where n is the number of dimensions.
One might ask, it possible to have quantum mechanics in space where the symmetry is a different group? Sp(n) and SU(n) have invariants of the form xy-yx, which unfortunately are identically zero for commuting coordinates. So really we are only interested in groups that can be defined in terms of symmetric invariants.
An interesting group is F4. In 26 dimensions it has the symmetric invariant (S):
So we could define a quantum theory in this space with an invariant based on these two invariants. For example we could define our amplitude as:
exp(iL + jS)
i X'X' + jX'X'X' ?
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