More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight.
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Here \ langle \ overline \ Psi \ Psi \ rangle is the scalar density of fermions, averaged on statistical ensemble:
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This has significance in applied mathematics and physics : if is some scalar density field and are the position vector coordinates, i . e . some measure theory.
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The reason is that the rescaled Hamiltonian constraint is a scalar density of weight two while it can be shown that only scalar densities of weight one have a chance to result in a well defined operator.
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The reason is that the rescaled Hamiltonian constraint is a scalar density of weight two while it can be shown that only scalar densities of weight one have a chance to result in a well defined operator.
6.
Notice that, in the presence of gravity or when using general curvilinear coordinates, the Lagrangian density \ mathcal { L } will include a factor of } } or its equivalent to ensure that it is a scalar density so that the integral will be invariant.
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