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strangerep

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energy eigenvector with non-zero eigenvalue?

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I think your reasoning is correct. If there is a reference frame in which energy E is non-zero and momentum is (supposedly) zero, then by boosting this state to various moving reference frames you'll obtain states with arbitrary momenta and energies E' greater than E. In relativistic QM this set of states is normally attributed to a particle with mass m = E/c^2. The unique vacuum state is normally associated with p=0, E=0.

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All representations of the Poincare group (= Lorentz group plus translations) that are relevant to physics are projective representations. There is a theorem (due to Bargmann, if I remember correctly) that projective representations of the Poincare group are equivalent to ordinary unitary representations. Another theorem (Wigner, 1939) provides a classification of the simplest (irreducible) unitary representations of the Poincare group. There are representations of three types (if we ignore spin), that can be classified according to allowed values of momentum p, energy E and mass m connected to each other by the usual relativistic formula

[tex] m = \frac{1}{c^2} \sqrt{E^2 - \mathbf{p}^2 c^2} [/tex]

These three types are normally associated with massive particles, massless particles, and vacuum:

massive particles: [itex] \mathbf{p} \in R^3; \ E > 0; \ m > 0 [/itex]

massless particles: [itex] \mathbf{p} \in R^3; \ E > 0; \ m = 0 [/itex]

vacuum: [itex] \mathbf{p} = 0; \ E = 0; \ m = 0 [/itex]

So, if the condition says explicitly that [itex] E \neq 0 [/itex], I don't see how it can be a vacuum in the usual sense.

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Haelfix

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Be careful, Supersymmetry is a space-time symmetry (it bypasses the Coleman Mandula theorem), so the relevant group is the Super Poincare group.

In fact with global supersymmetry you have identically zero canonical vacuum energy, given by the antibracket of the supercharges. However, SuSy must be broken at some scale. So instead we redo the calculation and parametrize our ignorance on the breaking mechanism (hence the degrees of freedom go up roughly by 100 in the mSSm). In some of the more popular ways of SuSy breaking (say in the context of SUGRA), you are left with the vacuum as a difference of two terms: The Kahler potential and a scalar field.

This is the cosmological constant problem. These two terms must delicately cancel to unbelievably precise accuracy, through some finetuning of 64 orders of magnitude worth (depending on how you count and as given by current experimental bounds) away from their natural values.

In fact with global supersymmetry you have identically zero canonical vacuum energy, given by the antibracket of the supercharges. However, SuSy must be broken at some scale. So instead we redo the calculation and parametrize our ignorance on the breaking mechanism (hence the degrees of freedom go up roughly by 100 in the mSSm). In some of the more popular ways of SuSy breaking (say in the context of SUGRA), you are left with the vacuum as a difference of two terms: The Kahler potential and a scalar field.

This is the cosmological constant problem. These two terms must delicately cancel to unbelievably precise accuracy, through some finetuning of 64 orders of magnitude worth (depending on how you count and as given by current experimental bounds) away from their natural values.

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