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Continuous Symmetries and Conservation Laws. Noether's Nature always creates the best of all options. Aristoteles (384 BC–322 BC) 8 Continuous Symmetries and Conservation Laws. Noether’s Theorem In many physical systems, the action is invariant under some continuous set of transformations. In such systems, there exist local and global conservation laws analogous to current and charge conservation in electrodynamics. The analogs of the charges can be used to generate the symmetry transformation, from which they were derived, with the help of Poisson brackets, or after quantization, with the help of commutators. 8.1 Point Mechanics Consider a simple mechanical system with a generic action tb A = dt L(q(t), q˙(t), t). (8.1) Zta 8.1.1 Continuous Symmetries and Conservation Law Suppose A is invariant under a continuous set of transformations of the dynamical variables: q(t) → q′(t)= f(q(t), q˙(t)), (8.2) where f(q(t), q˙(t)) is some functional of q(t). Such transformations are called sym- metry transformations. Thereby it is important that the equations of motion are not used when establishing the invariance of the action under (8.2). If the action is subjected successively to two symmetry transformations, the re- sult is again a symmetry transformation. Thus, symmetry transformations form a group called the symmetry group of the system. For infinitesimal symmetry trans- formations (8.2), the difference ′ δsq(t) ≡ q (t) − q(t) (8.3) will be called a symmetry variation. It has the general form δsq(t)= ǫ∆(q(t), q˙(t), t). (8.4) 619 620 8 Continuous Symmetries and Conservation Laws. Noether’s Theorem Symmetry variations must not be confused with ordinary variations δq(t) used in Section 1.1 to derive the Euler-Lagrange equations (1.8). While the ordinary vari- ations δq(t) vanish at initial and final times, δq(tb) = δq(ta) = 0 [recall (1.4)], the symmetry variations δsq(t) are usually nonzero at the ends. Let us calculate the change of the action under a symmetry variation (8.4). Using the chain rule of differentiation and an integration by parts, we obtain tb tb ∂L ∂L ∂L δsA = dt − ∂t δsq(t)+ δsq(t) . (8.5) ta "∂q(t) ∂q˙(t)# ∂q˙(t) Z ta For orbits q(t) that satisfy the Euler-Lagrange equations (1.8), only boundary terms survive, and we are left with t ∂L a δsA = ǫ ∆(q, q,˙ t) . (8.6) ∂q˙ tb Under the symmetry assumption, δsA vanishes for any orbit q(t), implying that the quantity ∂L Q(t) ≡ ∆(q, q,˙ t) (8.7) ∂q˙ is the same at times t = ta and t = tb. Since tb is arbitrary, Q(t) is independent of the time t, i.e., it satisfies Q(t) ≡ Q. (8.8) It is a conserved quantity, a constant of motion. The expression on the right-hand side of (8.7) is called Noether charge. The statement can be generalized to transformations δsq(t) for which the action is not directly invariant but its symmetry variation is equal to an arbitrary boundary term: tb δsA = ǫ Λ(q, q,˙ t) . (8.9) ta In this case, ∂L Q(t)= ∆(q, q,˙ t) − Λ(q, q,˙ t) (8.10) ∂q˙ is a conserved Noether charge. It is also possible to derive the constant of motion (8.10) without invoking the action, but starting from the Lagrangian. For it we evaluate the symmetry variation as follows: ∂L ∂L d ∂L δsL≡L (q+δsq, q˙+δsq˙) − L(q, q˙)= −∂t δsq(t)+ δsq(t) . (8.11) "∂q(t) ∂q˙(t)# dt "∂q˙(t) # 8.1 Point Mechanics 621 On account of the Euler-Lagrange equations (1.8), the first term on the right-hand side vanishes as before, and only the last term survives. The assumption of invariance of the action up to a possible surface term in Eq. (8.9) is equivalent to assuming that the symmetry variation of the Lagrangian is a total time derivative of some function Λ(q, q,˙ t): d δ L(q, q,˙ t)= ǫ Λ(q, q,˙ t). (8.12) s dt Inserting this into the left-hand side of (8.11), we find d ∂L ǫ ∆(q, q,˙ t) − Λ(q, q,˙ t) =0, (8.13) dt " ∂q˙ # thus recovering again the conserved Noether charge (8.8). The existence of a conserved quantity for every continuous symmetry is the content of Noether’s theorem [1]. 8.1.2 Alternative Derivation Let us do the substantial variation in Eq. (8.5) explicitly, and change a classical orbit qc(t), that extremizes the action, by an arbitrary variation δaq(t). If this does not vanish at the boundaries, the action changes by a pure boundary term that follows directly from (8.5): t ∂L b δaA = δaq . (8.14) ∂q˙ ta From this equation we can derive Noether’s theorem in yet another way. Suppose we subject a classical orbit to a new type of symmetry variation, to be called local symmetry transformations, which generalizes the previous symmetry variations (8.4) by making the parameter ǫ time-dependent: t δsq(t)= ǫ(t)∆(q(t), q˙(t), t). (8.15) t The superscript t of δsq(t) indicates the new time dependence in the parameter ǫ(t). These variations may be considered as a special set of the general variations δaq(t) t introduced above. Thus also δsA must be a pure boundary term of the type (8.14). For the subsequent discussion it is useful to introduce the infinitesimally transformed orbit ǫ t q (t) ≡ q(t)+ δsq(t)= q(t)+ ǫ(t)∆(q(t), q˙(t), t), (8.16) and the associated Lagrangian: Lǫ ≡ L(qǫ(t), q˙ǫ(t)). (8.17) Using the time-dependent parameter ǫ(t), the local symmetry variation of the action can be written as tb tb ǫ ǫ ǫ t ∂L d ∂L d ∂L δsA = dt − ǫ(t)+ ǫ(t) . (8.18) ta "∂ǫ(t) dt ∂ǫ˙(t)# dt " ∂ǫ˙ # Z ta 622 8 Continuous Symmetries and Conservation Laws. Noether’s Theorem Along the classical orbits, the action is extremal and satisfies the equation δA =0, (8.19) δǫ(t) which translates for a local action to an Euler-Lagrange type of equation: ∂Lǫ d ∂Lǫ − =0. (8.20) ∂ǫ(t) dt ∂ǫ˙(t) This can also be checked explicitly by differentiating (8.17) according to the chain rule of differentiation: ∂Lǫ ∂Lǫ ∂Lǫ = ∆(q, q,˙ t)+ ∆(˙ q, q,˙ t); (8.21) ∂ǫ(t) ∂q(t) ∂q˙(t) ∂Lǫ ∂Lǫ = ∆(q, q,˙ t), (8.22) ∂ǫ˙(t) ∂q˙(t) and inserting on the right-hand side the ordinary Euler-Lagrange equations (1.8). We now invoke the symmetry assumption that the action is a pure surface term under the time-independent transformations (8.15). This implies that ∂Lǫ d = Λ. (8.23) ∂ǫ dt Combining this with (8.20), we derive a conservation law for the charge: ∂Lǫ Q = − Λ. (8.24) ∂ǫ˙ Inserting here Eq. (8.22), we find that this is the same charge as that derived by the previous method. 8.2 Displacement and Energy Conservation As a simple but physically important example consider the case that the Lagrangian does not depend explicitly on time, i.e., that L(q, q,˙ t) ≡ L(q, q˙). Let us perform a time translation on the coordinate frame: t′ = t − ǫ. (8.25) In the new coordinate frame, the same orbit has the new description q˙(t′)= q(t), (8.26) i.e., the orbitq ˙(t) at the translated time t′ is precisely the same as the orbit q(t) at the original time t. If we replace the argument ofq ˙(t) in (8.26) by t′, we describe a 8.2 Displacement and Energy Conservation 623 time-translated orbit in terms of the original coordinates. This implies the symmetry variation of the form (8.4): ′ ′ δsq(t) = q (t) − q(t)= q(t + ǫ) − q(t) = q(t′)+ ǫq˙(t′) − q(t)= ǫq˙(t). (8.27) The symmetry variation of the Lagrangian is in general ∂L ∂L δ L = L(q′(t), q˙′(t)) − L(q(t), q˙(t)) = δ q(t)+ δ q˙(t). (8.28) s ∂q s ∂q˙ s Inserting δsq(t) from (8.27) we find, without using the Euler-Lagrange equation, ∂L ∂L d δsL = ǫ q˙ + q¨ = ǫ L. (8.29) ∂q˙ ∂q˙ ! dt This has precisely the form of Eq. (8.12), with Λ = L as expected, since time translations are symmetry transformations. Here the function Λ in (8.12) happens to coincide with the Lagrangian. According to Eq. (8.10), we find the Noether charge ∂L Q = q˙ − L(q, q˙) (8.30) ∂q˙ to be a constant of motion. This is recognized as the Legendre transform of the Lagrangian which is, of course, the Hamiltonian of the system. s Let us briefly check how this Noether charge is obtained from the alternative formula (8.10). The time-dependent symmetry variation is here t δsq(t) = ǫ(t)q ˙(t), (8.31) under which the Lagrangian is changed by ∂L ∂L ∂Lǫ ∂Lǫ δtL = ǫq˙ + (˙ǫq˙ + ǫq¨)= ǫ + ǫ,˙ (8.32) s ∂q ∂q˙ ∂ǫ˙ ∂ǫ˙ with ∂Lǫ ∂L = q˙ (8.33) ∂ǫ˙ ∂q˙ and ∂Lǫ ∂L ∂L d = q˙ + ǫq¨ = L. (8.34) ∂ǫ ∂q ∂q˙ dt This shows that time translations fulfill the symmetry condition (8.23), and that the Noether charge (8.24) coincides with the Hamiltonian found in Eq. (8.10). 624 8 Continuous Symmetries and Conservation Laws. Noether’s Theorem 8.3 Momentum and Angular Momentum While the conservation law of energy follows from the symmetry of the action under time translations, conservation laws of momentum and angular momentum are found if the action is invariant under translations and rotations.
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