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.
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Notes
- 1.
Latin indices from the middle of the alphabet, i, j, k, are reserved for those coordinates whose velocities can be found from (8.29). Greek indices from the beginning of the alphabet, α, β, γ, are used to denote the remaining coordinates.
- 2.
Indeed, if they depended on one of \(\dot q^\alpha\), it would be possible to find it in terms of q, p, contradicting the rank condition (8.28).
- 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.
If not, the initial variables q A can be re-numbered to achieve this.
- 5.
The velocities v α which “survive” after the variable change are just the Lagrangian multipliers of the Dirac recipe, see below.
- 6.
Remember that the secondary constraints consist of the second-stage, third-stage, … constraints.
- 7.
Recall 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.
- 8.
Remember that all the variables \(x^\mu(\tau)\), used for the description of the relativistic particle, have no direct physical meaning. The same can happen in the general case. Generally, neither \(z^A\) nor any part of them represent physically measurable positions and velocities.
- 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. (8.116).
- 10.
We do not discuss the problem of ordering of operators which must be solved in each concrete case.
- 11.
Poisson bracket in field theory is defined by \(\{A(x), B(y)\}\!=\!\int d^3z\left[\frac{\delta A(x)}{\delta\phi^A(z)}\frac{\delta B(y)}{\delta p_A(z)}\right.\) \(\left.-\frac{\delta A(x)}{\delta p_A(z)}\frac{\delta B(y)}{\delta\phi^A(z)}\right]\). A and B are taken at the same instance of time. The working formula for computing the variational derivative is \(\frac{\delta A(\phi(x), \partial_b\phi(x))}{\delta\phi^A(z)}\!=\!\left.\frac{\partial A}{\partial\phi^A}\right|_{\phi\rightarrow\phi(x)}\delta^3(x-z)\!+\!\left.\frac{\partial A}{\partial\partial_b\phi^A}\right|_{\phi\rightarrow\phi(x)}\frac{\!\partial}{\!\partial xb}\!\delta^3(x-z)\).
- 12.
There is an elegant formalism developed by Berezin and Marinov [42] based on using anticommuting variables for semi-classical description of spin. We present here a formulation based on commuting variables, without appealing to the rather formal methods of Grassmann mechanics.
- 13.
It is well known that these six variables can be used to construct the quantities, \(L_i=\epsilon_{ijk}\omega_j\pi_k\), which obey the angular-momentum algebra. The problem is that, according to (8.163), we need a space with two degrees of freedom instead of six.
- 14.
We point out that when \(a^2=1\), it is precisely the Lorentz-group algebra written in terms of the rotation J and the Lorentz boost \(\tilde\pi\) generators.
- 15.
\(SO(3)\) -algebra has only one Casimir operator J 2.
- 16.
The first two terms resemble the relativistic particle Lagrangian \(\frac{1}{2e}\dot x^2+\frac{em^2}{2}\), the last one implies \(p^2=m^2\).
- 17.
More exactly, the first-class constraint is given by the combination \(\pi^2-\frac{3\hbar^2}{4a^2} + \frac{3\hbar^2}{4a^4}(v^2-a^2) = 0\).
- 18.
As will be shown below, Eq. (8.234) 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.
- 19.
In the transition from mechanics to a field theory, derivatives are replaced by variational derivatives. In particular, the last term in Eq. (8.235) reads \(\frac{\delta}{\delta\omega_i(x)} \int d^3ys^a(x)T_a(q^A(y), \omega_i(y)\).
- 20.
Here the condition (8.312) 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. 2.12. So the extra constraints would be responsible for ruling out these hidden degrees of freedom. Our condition (8.312) precludes the appearance of the hidden degrees of freedom.
- 21.
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].
- 22.
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: \(a\rightarrow a^{'}=\alpha a, \ b\rightarrow b^{'}=\alpha^{-1}b\). Another simple possibility is to write \(q=a+b\), which implies the symmetry \(a\rightarrow a^{'}=a+\alpha, \ b\rightarrow b^{'}=b-\alpha\). The well-known example of this kind transformation is einbein formulation in gravity theory: \(g_{\mu\nu}=e^a_\mu e^a_\nu\), which implies local Lorentz invariance.
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Deriglazov, A. (2010). Hamiltonian Formalism for Singular Theories. In: Classical Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14037-2_8
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