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Hamiltonian Formalism for Singular Theories

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Abstract

Modern particle and field theories often involve auxiliary variables which have no direct physical meaning. We have seen examples of this kind at the end of first chapter: Lagrangian multipliers for holonomic constraints, forceless Hertz mechanics, electrodynamics and the relativistic particle. Their auxiliary character is supplied either by local symmetries presented in the Lagrangian action, or by the algebraic character of equations of motion for these variables. So, in Lagrangian formalism we deal with a singular theory. Equations of motion can have rather a complicated structure, including in general differential equations of the second and the first order as well as algebraic equations. Besides, identities among the equations can be present in the formulation. As a consequence, there is an ambiguity in solutions for any given initial conditions. So, the physical content of a theory with local symmetry is not a simple question. Hamiltonian formalism is well adapted for the investigation of a singular theory.

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Notes

  1. 1.

    Latin indices from the middle of the alphabet, i, j, k, are reserved for those coordinates whose velocities can be found from (8.31). Greek indices from the beginning of the alphabet, α, β, γ, are used to denote the remaining coordinates.

  2. 2.

    Indeed, if they depended on one of \(\dot{q}^{\alpha }\), it would be possible to find it in terms of others q and p, contradicting the rank condition (8.30).

  3. 3.

    Note that we can substitute the velocities \(v^{\underline{\alpha }}(z,v^{\bar{\beta }})\) into the complete Hamiltonian before computing the Poisson brackets, which does not alter the resulting equations of motion. This follows from the fact that the velocities enter into H multiplied by the primary constraints. So, on the constraint surface, \(\{z,v^{\alpha '}(z)\Phi _{\alpha }\} =\{ z,\Phi _{\alpha }\}v^{\alpha '}(z) +\{ z,v^{\alpha '}(z)\}\Phi _{\alpha } =\{ z,\Phi _{\alpha }\}v^{\alpha '}\).

  4. 4.

    If not, the initial variables q A can be re-numbered to achieve this.

  5. 5.

    The velocities v α which “survive” after the variable change are just the Lagrangian multipliers of the Dirac recipe, see below.

  6. 6.

    Remember that the secondary constraints consist of the second-stage, third-stage, …constraints.

  7. 7.

    Recall that we can substitute the velocities \(v^{\underline{\alpha }}(z,\bar{v}^{\bar{\beta }})\) into the complete Hamiltonian before computing the Poisson brackets, which does not alter the resulting equations of motion.

  8. 8.

    For a detailed description of the class, see [10].

  9. 9.

    The operators \(\hat{q}\), \(\hat{p}\) are taken as hermitian, which guarantees that their eigenvalues are real numbers. Since the commutator of Hermitian operators is an anti-Hermitian operator, the factor i appears on the r.h.s. of Eq. (8.117).

  10. 10.

    We do not discuss the problem of ordering of operators which must be solved in each concrete case.

  11. 11.

    We point out that, contrary to electrodynamics, each class of equivalent configurations, \(\tilde{x} =\{ (cf(\tau ),\mathbf{x}(\,f(\tau ))),\mathbf{x}\,\mbox{ is a given function,}\,\frac{df} {d\tau } > 0\}\), contains a representative which is an observable quantity: x μ(τ) = (c τ, x(τ)).

  12. 12.

    This is not a surprise, since we already noticed that (8.189) is a consequence of (8.190).

  13. 13.

    Poisson bracket in field theory is defined by \(\{A(x),B(y)\} =\int d^{3}z\left [\frac{\delta A(x)} {\delta \phi ^{A}(z)} \frac{\delta B(y)} {\delta p_{A}(z)}\right.\left.-\frac{\delta A(x)} {\delta p_{A}(z)} \frac{\delta B(y)} {\delta \phi ^{A}(z)} \right ]\). A and B are taken at the same instance of time. The working formula for computing the variational derivative is \(\frac{\delta A(\phi (x),\partial _{b}\phi (x))} {\delta \phi ^{A}(z)} = \left.\frac{\partial A} {\partial \phi ^{A}}\right \vert _{\phi \rightarrow \phi (x)}\delta ^{3}(x - z) + \left. \frac{\partial A} {\partial \partial _{b}\phi ^{A}}\right \vert _{\phi \rightarrow \phi (x)} \frac{\partial } {\partial xb}\delta ^{3}(x - z)\).

  14. 14.

    After rescaling the parameter, \(\epsilon = -\sqrt{-\dot{x}^{2}}\epsilon '\), it acquires the standard form of a reparametrization.

  15. 15.

    As will be shown below, Eq. (8.275) represents a solution to the equation \(\tilde{p} _{j} = \frac{\partial \tilde{L} } {\partial \dot{q}^{j}}\) defining the conjugate momenta \(\tilde{p} _{j}\) of the extended formulation.

  16. 16.

    In the transition from mechanics to a field theory, derivatives are replaced by variational derivatives. In particular, the last term in Eq. (8.276) reads \(\frac{\delta }{\delta \omega _{ i}(x)}\int d^{3}ys^{a}(x)T_{a}(q^{A}(y),\omega _{i}(y)\).

  17. 17.

    Here the condition (8.354) is important. A theory with higher derivatives, being equivalent to the initial one, has more degrees of freedom than the number of variables q A, see Sect. 2.10 So the extra constraints would be responsible for ruling out these hidden degrees of freedom. Our condition (8.354) precludes the appearance of the hidden degrees of freedom.

  18. 18.

    This is a general situation: given a locally-invariant action, there are special coordinates such that the action does not depend on some of them [10].

  19. 19.

    There are other possibilities for creating trivial local symmetries. For example, in a given Lagrangian action with one of the variables being q, let us make the substitution q = ab, where a, b represent new configuration space variables. The resulting action is equivalent to the initial one, an auxiliary character of one of the new degrees of freedom is guaranteed by the trivial gauge symmetry: a → a′ = α a,  b → b′ = α −1 b. Another simple possibility is to write q = a + b, which implies the symmetry a → a′ = a +α,  b → b′ = b −α. The well-known example of this kind transformation is einbein formulation in gravity theory: g μ ν  = e μ a e ν a, which implies local Lorentz invariance.

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Authors and Affiliations

  1. Universidade Federal de Juiz de Fora Dept. of Mathematics, Juiz de Fora, Minas Gerais, Brazil

    Alexei Deriglazov

  2. Laboratory of Mathematical Physics, Tomsk Polytechnic University, Tomsk, Russian Federation

    Alexei Deriglazov

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Deriglazov, A. (2017). Hamiltonian Formalism for Singular Theories. In: Classical Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-44147-4_8

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