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I met the word smearing function (or test function) when I was learning ADM formalism in GR books. What makes me scratch my head is the reason of introducing such a smearing function when we calculate the Poisson bracket of constraints.

For example, when we are calculating time evolution of a primary constraint $\phi^a=0$, we should in principle have $$\int s_a\partial_t\phi^a d^3x=\int s_a\left[\phi^a,\mathcal{H}\right] d^3x\simeq0$$ where $\mathcal{H}$ is the Hamiltonian density, "$\simeq$" simbol means that this equality meets on shell, and $s_a$ is the corresponding smearing function which goes to $0$ very fast on the boundary and is $C^\infty$ (infinitely differentiable)

My question is then,

  1. why we shall introduce such a smearing function rather than do the Poisson bracket directly?
  2. suppose two another smearing functions $m^a$, $n^b$, and an existing vector field depending on space and time $\psi^a(\textbf{x},t)$, which is also $C^\infty$ and vanishes on the boundary. We have $$m^an^b=\psi^ap^b$$ $$m^a\psi^b=n^aq^b$$ Here, may I ask whether $p^b$ and $q^b$ are still smearing functions?

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Let me give an example to the second question. Suppose we have a primary constraint $T_{ab}\psi^a=0$. Here $T_{ab}$ is a tensor and $\psi=\psi^a(\textbf{x},t)$ is the vector field mentioned above. Now, when we are calculating the secondary constraints, since calculating Poisson bracket makes us have two smearing function, we find, let's say, such a term $$\int m^an^bT_{ab}d^3x$$ could we now use $m^an^b=\psi^ap^b$ so that this term becomes $$\int m^an^bT_{ab}d^3x=\int\psi^ap^bT_{ab}d^3x$$ and thus vanishes because of the primary constraint?
Thanks for the attention

Chunhui
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1 Answers1

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In this answer we address OP's first question:

  1. In Hamiltonian field theory, in order to avoid multiplying two distributions together, the Poisson bracket is defined for local functional (rather than point-wise), cf. e.g. my Phys.SE answer here. We therefore turn the constraint $\phi^a(x)$ into a local functional $$\Phi^a[s]~=~\int_Mdx~ s(x)\phi^a(x),$$ where $s(x)$ is a test function. So instead of saying that we have a constraint for each point $x\in M$, we have a constraint for each test function $s$, which is a more well-defined/smooth way of implementing the constraint.
Qmechanic
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