| dbp:proof
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- In standard single-variable calculus, the derivative of a smooth function f is defined by
This can be rearranged to find f:
It follows that is a translation operator. This is instantly generalized to multivariable functions f
Here is the directional derivative along the infinitesimal displacement ε. We have found the infinitesimal version of the translation operator:
It is evident that the group multiplication law U'U=U takes the form
So suppose that we take the finite displacement λ and divide it into N parts , so that λ/N=ε. In other words,
Then by applying U N times, we can construct U:
We can now plug in our above expression for U:
Using the identity
we have
And since we have
Q.E.D.
As a technical note, this procedure is only possible because the translation group forms an Abelian subgroup in the Poincaré algebra. In particular, the group multiplication law U'U = U should not be taken for granted. We also note that Poincaré is a connected Lie group. It is a group of transformations T that are described by a continuous set of real parameters . The group multiplication law takes the form
Taking as the coordinates of the identity, we must have
The actual operators on the Hilbert space are represented by unitary operators U. In the above notation we suppressed the T; we now write U as U. For a small neighborhood around the identity, the power series representation
is quite good. Suppose that U form a non-projective representation, i.e.,
The expansion of f to second power is
After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition
Since is by definition symmetric in its indices, we have the standard Lie algebra commutator:
with C the structure constant. The generators for translations are partial derivative operators, which commute:
This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means that f is simply additive:
and thus for abelian groups,
Q.E.D. (en)
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