Abstract
It was mentioned in Sect. 2.5 that conservation laws play an important role in the analysis of classical and quantum systems. This chapter is mainly devoted to discussion of the Noether theorem, which gives the relationship between the existence of conservation laws for the system in question, and symmetries of the associated action functional. The symmetries usually have a certain physical interpretation; in particular, they may reflect some fundamental properties assumed for our space-time: homogeneity, isotropy, … . In this case, the Noether theorem states that conservation laws are consequences of these properties. For example, symmetry under spatial translations implies the conservation of the total momentum of a system.
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Notes
- 1.
For a free particle, solutions to equations of motion \(x^a(x^0)\) are linear functions, and Eq. (7.47) can be resolved; see Example 1. Serious problems arise for particle and field theories with interaction.
- 2.
See [37] for discussion of the most general form of the Noether theorem.
- 3.
As we will see below, D determines an infinitesimal transformation of a function.
- 4.
Note that in singular theory it can happen that \(Q\equiv 0\), which implies identities among the equations of motion. This is closely related with the presence of local symmetries, see Chap. 8.
- 5.
It is reasonable to divide by \(m_1+m_2\), then X has the dimension of a length.
- 6.
Of course, the problem can easily be solved. We write \(\omega^{ij}x^j \equiv \frac12(\delta^{ik}x^j-\delta^{ij}x^k)\omega^{kj}\). Then the quantities \(R^i_{kj} = \frac12(\delta^{ik}x^j-\delta^{ij}x^k)\) with \(k<j\) represent the generators.
- 7.
One cannot compute the bracket \(\{z^i, Q\}\) directly, since Q contains the unspecified function N.
References
P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, NY, 1986)
J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill Book Company, New York, NY, 1964)
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Deriglazov, A. (2010). Transformations, Symmetries and Noether Theorem. In: Classical Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14037-2_7
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