Directional derivative
Translations

In the Poincaré algebra, we can define an infinitesimal translation operator P as

(the i ensures that P is a self-adjoint operator) For a finite displacement λ, the unitary Hilbert space representation for translations is

By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:

This is a translation operator in the sense that it acts on multivariable functions f (x) as

In standard single-variable calculus, the derivative of a smooth function f (x) is defined by (for small ε)

This can be rearranged to find f (x + ε):

It follows that {\displaystyle } is a translation operator. This is instantly generalized to multivariable functions f (x)

Here ε ⋅ ∇ {\displaystyle {\boldsymbol {\varepsilon }}\cdot \nabla } 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 (g) U (f)= U (gf) takes the form

So suppose that we take the finite displacement λ and divide it into N parts (N →∞ is implied everywhere), so that λ / N = ε. In other words,

Then by applying U (ε) N times, we can construct U (λ):

We can now plug in our above expression for U(ε):

Using the identity

we have

And since U (ε) f (x) = f (x + ε) we have

Q.E.D.

As a technical note, this procedure is only possible because the translation group forms an Abelian subgroup (Cartan subalgebra) in the Poincaré algebra. In particular, the group multiplication law U (a) U (b) = U (a + b) should not be taken for granted. We also note that Poincaré is a connected Lie group. It is a group of transformations T (ξ) that are described by a continuous set of real parameters ξ a {\displaystyle \xi ^{a}}. The group multiplication law takes the form

Taking ξ a = 0 {\displaystyle \xi ^{a}=0} as the coordinates of the identity, we must have

The actual operators on the Hilbert space are represented by unitary operators U (T (ξ)). In the above notation we suppressed the T; we now write U (λ) as U (P (λ)). For a small neighborhood around the identity, the power series representation

is quite good. Suppose that U(T(ξ)) 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 t a b {\displaystyle t_{ab}} 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.
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