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I'm performing a calculation involving Dirac constraints theory, and I need to calculate the Poisson brackets between constraints and the total Hamiltonian. The starting theory is described by a Lagrangian density depending on two fields (say, L(X,Y)) and I realized I'm not familiar with a definition of Poisson brackets for a field theory involving more than a single field. I wasn't able to find a book that generalizes the standard definition for discrete systems (https://en.wikipedia.org/wiki/Poisson_bracket) to fields, so my first question is a good book for reference. Secondly, by thinking by analogy I came up with two different ways to solve my problem:

  1. I define a Poisson bracket for every couple of field-momentum: \begin{equation} \left\{ f,g \right\}= \frac{\partial f}{\partial X}\frac{\partial g}{\partial V}-\frac{\partial f}{\partial V}\frac{\partial g}{\partial X} \end{equation} for the PB with respect to the field X and the corresponding momentum V, and: \begin{equation} \left\{ f,g \right\}= \frac{\partial f}{\partial Y}\frac{\partial g}{\partial W}-\frac{\partial f}{\partial W}\frac{\partial g}{\partial Y} \end{equation} for the PB with respect to the field Y and the corresponding momentum W
  2. I define a single PB summing up for all the field-momentum couples (similarly to che standard definition):

\begin{equation} \left\{ f,g \right\}= \frac{\partial f}{\partial X}\frac{\partial g}{\partial V}-\frac{\partial f}{\partial V}\frac{\partial g}{\partial X} + \frac{\partial f}{\partial Y}\frac{\partial g}{\partial W}-\frac{\partial f}{\partial W}\frac{\partial g}{\partial Y} \end{equation}

Can someone help?

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1 Answer 1

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I can't help you with references, but there are lots of works on classical canonical transformations in classical field theory.

It is standard for one pair of conjugate fields $X(x)$ and $V(x)$ and a functional of them, $f[X,V]$, to consider
$$ \left\{ f[X,Y] , g[X,Y]\right \}= \int\!\! dx ~~~\Bigl ( \frac{\delta f}{\delta X(x)}\frac{\delta g}{\delta V(x)}- \frac{\delta f}{\delta V(x)}\frac{\delta g}{\delta X(x)} \Bigr ), $$ generalizing the standard multivariable expression to an infinite vector of variables, the fields.

If you have several fields, you may index fields to $X^i(x)$ and $V^i(x)$, and you further multiplex the above expression to discrete vectors whose components are infinite-dimensional fields, $$ \left\{ f[X,Y] , g[X,Y]\right \}=\sum_i \int\!\! dx ~~~\Bigl ( \frac{\delta f}{\delta X^i(x)}\frac{\delta g}{\delta V^i(x)}- \frac{\delta f}{\delta V^i(x)}\frac{\delta g}{\delta X^i(x)} \Bigr ). $$

Recall $$ \left\{ X^k(z) , V^j(y)\right \}= \delta^{kj} \delta (z-y)~~. $$

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