Noether s Theorem: Uses and Abuses

Transcription

1 Noether s Theorem: Uses and Abuses Ryan Browne December 15, 2011 Contents 1 Introduction 1 2 Formulation Definitions Formal Statement of Noether s Theorem Form of the conserved quantities Example Gauge Theories Gauge Theories Caveat Emptor Lagrangian Wanted Not Quite a Conservation Law Inequivalent Lagrangians Other Conservation Laws Dynamical Conservations Laws Solitons Conclusion 9 1 Introduction Proven in 1915 and published in 1918, Noether s theorem, as applied to physics, tells us about conservation laws and conserved quantities in dynam- 1

2 ical systems. In 1915, general relativity was almost a finished theory, but there was a problem regarding the conservation of energy. Hilbert had noted that the failure of general relativity to produce a classical conservation of energy equation was intrinsic to the system. He asked Noether to help clarify his idea. Noether proceeded to derive two foundational theorems which confirmed Hilbert s suspicion and fundamentally changed the way that the Lagrangian formalism was seen. [4] Due to its generality, Noether s theorem has seen use in every aspect of physics and has gained great noteriety amoungst physicists. However, there are several key features of Noether s theorem which limit its applicability to physical systems, especially when compared to the popular notion of the theorem. It is often said that Noether s theorem related conservation laws and symmetries of the action of a system, and, while this is based in truth, it is not true in general. In this paper, we will explore Noether s theorem to categorize the areas in which it is and is not applicapble. 2 Formulation In researching this paper, I found a near infinite supply of different ways of stating Noether s theorem. Many of them, it turns out are not mathematically rigorous, and can lead to misinterpretation and misapplication of the theorem. Therefore, we will go back to Noether s 1918 paper to find the theorem in its true form. Before stating Noether s theorem, it is important to understand the mathematical tools which are used. Noether did much of her work in abstract algebra, and her theorem relies on the tools available therein. Therefore, a brief introduction to abstract algebra, specifically group theory, is important. The fundamental idea to take away from this mathematical diversion is the idea of a Lie group, as Noether s theorem relies on it. 2.1 Definitions A group, in the mathematical sense, refers to a set of elements and the rules for composing those elements, called an operation. To qualify as a group, a set and its associated operation must meet the following requirements: the operation must be associative, the result of composing two elements must be an element of the group, the group must contain an identity element, and 2

3 the group must contain the inverse of every element in the group. These restrictions are written out explicitly below, where the group is G and the operation is multiplication. [8] Associativity : a (b c) = (a b) c a, b, c G Closure : a b = c c G a, b G Identity : I a = a I G a G Inverse : a a 1 = I a 1 G a G A symmetry group is a property of an object and is the group of transformations of that object which do not change the object. An equilateral triangle looks the same if rotated by 2 n π radians. Thus it has a symmetry 3 group which contains the discrete rotations by 2 n π radians where n I. 3 Similarly, a uniform sphere, regardless of its angular orientation, looks the same; it is invariant under all rotations about its center. Therefore, it has a rotational symmetry group containing the continuous set of rotations about its center. [9] In the example of the triangle, the angle of rotation can only be certain, discrete values, so it is called a discrete symmetry group. Correspondingly, the symmetry group of rotations of the sphere is a continuous group, as the angle by which the sphere can be rotated, and still look the same can be any real number. [9] A Lie group is a continuous, smooth symmetry group. For a symmetry group to be a Lie group, the equations which define its transformation must be differentiable. [7] 3

4 2.2 Formal Statement of Noether s Theorem Given a Lie group G, whose most general transform depends on ρ parameters, under the action of which an integral I is invariant, there are ρ linearly independent combinations of the Lagrange expressions which become divergences. The converse also holds true; the existence of ρ Lagrange expressions which are divergences implies invariance of I under the action of a transform in G.[6] The Lagrange expressions which are referenced in the theorem are the left side of the Euler-Lagrange equations, what most physicists ( ) think of when dl they think of the Euler-Lagrange equations, i.e. dx i d dl dx i dx i. [6] Note that the right side of the Euler-Lagrange equations is 0 when the system satisfies Hamilton s principle. While this result is not used in deriving Noether s theorem, it will come in later. As an example, consider the action of a one-dimensional Lagrangian density of the form dxl(φ(x), dφ, x), which is invariant under translations along dx the real number line. For that action, Noether s theorem states that the derivative is a divergence, i.e. Γ = dl dφ d ( ) dl dx d dφ dx Γ = dρ (2) dx Remember that the number of divergences which Noether s theorem produces is affected only by the symmetry group being investigated, not the dimensionality of the action. Therefore, we can easily extend eqns (1,2) to four dimensions, if we hold the symmetry group fixed. This gives us Γ = dl dφ d dl i dx i d dφ (3) dx i Γ = dρ0 + ρ (4) dt where we use the normal expression for the four-vector ρ µ = {ρ 0, ρ}. Noether s theorem is often said to connect symmetries of the action to the conservation laws, but, all we have is that we can re-write the Euler-Lagrange equations in terms of divergences. However, we have not used Hamilton s Principle. The applicability of Hamilton s principle should be fairly obvious. 4 (1)

5 If the system satisfies Hamilton s Principle, then the Lagrange expressions are equal to zero. This means that dρ0 + ρ = 0. In the four-dimensional dt case, this gives us a continuity equation, which can be inverted to find a conserved charge. [6] 2.3 Form of the conserved quantities It is useful to know that, given a particular symmetry of the action of a system, there exist conservation laws. However, Noether s theorem goes even further. Noether derived the form of the conserved quantities, using the calculus of variations. The form of ρ is ρ µ i = k ( ) L ψ k ( dφ Λµ (5) k ) x µ x µ dx µ Where ρ µ i is the argument of the ith Noether divergence, φ k is a field of the Lagrangian density or a function of the Lagrangian, which can depend on all of the independent variables, ψ k is from the transformation of the field φ k φ k = φ k + ψ k under which the action is symmetric. ψ k can depend on all of the independent variables, and the number of parameters it has, sets the number of Noether divergences. Λ µ is a term which allows the Lagrangian to vary by up to a total derivative; Noether s theorem still holds under this transformation. [3] 2.4 Example Now, we can explore a quick example of Noether s theorem in action in physics. Consider the Lagrangian which gives rise to Schoedinger s equation. L = 2 2m ψ ψ + i 2 (ψ ψ ψ ψ) (6) In this Lagrangian density, ψ and ψ are consider independent functions. [3] The action of this Lagrangian density is symmetric under the transformation ψ ψ = ψ + ɛ where ɛ = ɛ R + iɛ I. Because this is still a one dimensional symmetry, there is only one conserved current. Also, Λ is zero 5

6 for this system. [3]Using Noether s theorem, ρ µ = ( ) L ɛ k k ( dψ k ) x µ dx ( µ ) ( ) L ρ µ ɛ L ɛ = ( dψ + ) x µ ( dψ ) x µ dx µ dx ( µ ) ρ = {i ψ, 2 2m ψ } + {i ψ, 2 m ψ} (7) (8) (9) The continuity equation of this gives dρ 0 ( ) dt + ρ = i ψ + 2 2m 2 ψ + i ψ + 2 m 2 ψ = 0 (10) This gives the Schoedinger equation of a free particle. [3] Using Gauss s Law, we can find that 3 Gauge Theories 3.1 Gauge Theories 2 Q = 2m ( ψ + ψ ) dx (11) It is important to note that Noether s theorem, as defined in this paper, cannot be used in any gauge theory. Gauge theories require that the Lagrangian be invariant under a local gauge transformation. Noether s theorem is only defined for systems which undergo global transformations. Global transformations affect every part of the system in the same way. For instance, if a system undergos a transformation where x x + δx, then every particle, indexed by i, which was defined by a position y i = x + ψ i will now be defined by a position y i = (x + δx) + ψ i. [1] A local transformation can affect every part of the system differently. Now, rather than x x + δx, the transformation is x x + f(x). In the language of Noether s theorem, rather than the general transformation depending on ρ parameters δx i, the general transformation will depend on ρ functions f i (x). To deal with this type of transformation, you need to use the second theorem presented in Noether s 1918 paper. [1] 6

7 Noether s second theorem is the result which confirmed Hilbert s thoughts on energy conservation in general relativity. The result that she derived, in a simplified form that is applicable to physical systems, provides another form of the conservation laws. However, the conservation laws derived from Noether s second theorem are dependent on the metric (or shape) of the space that is being analyzed, and a conservation law derived using one set of coordinates cannot be used in another set of coordinates. In this scenario, a law of conservation of momentum, derived in cartesian coordinates, would not be applicable to a system described by polar coordinates. [1] 4 Caveat Emptor 4.1 Lagrangian Wanted Noether s theorem, when applied to physics, requires an action to be defined for a system in order to say anything about the system. Thus, in systems which do not have a Lagrangian, Noether s theorem tells us nothing about it. One such system was put forward by Wigner to show the limitations of Noether s theorem in its applications to physics.[2] 4.2 Not Quite a Conservation Law Noether s theorem is usually referenced in relation to conservation laws, however, it is important to note that nothing about Noether s theorem guarantees that a symmetry of the action will produce a meaningful conservation law. In cases where there is not a meaningful conservation law, the continuity equation will exist, as defined by Noether s theorem, but the the boundary conditions may not permit a conversion from the continuity equations to an actual, conserved charge. The continuity equations require that the fields vanish at infinity, which allows the continuity equation to be inverted. [2] 4.3 Inequivalent Lagrangians When a system can be modeled by multiple Lagrangians that differ by more than a divergence term, the system is said to have inequivalent Lagrangians. The action associated with each Lagrangian will have its own set of symmetries. Those symmetries may correspond to different conservation laws in the 7

8 equations of motion, may correspond to the same conservation law, or some of the actions may have no symmetries at all. In order to determine all of the Noether currents of a system, all of the inequivalent Lagrangians for that system must be investigated.[5, 2] One example of this is can be seen in the action of the inequivalent Lagrangians for the two-dimensional harmonic oscillator. Two inequivalent Lagrangians can be derived for the system or L = 1 2 ( q q 2 2 ) ω 2 (q1 2 + q2) 2 (12) L = q 1 q 2 ω 2 q 1 q 2 (13) The action of the first Lagrangian is symmetric under 2-dimensional rotations, but the action of the second Lagrangian is symmetric under the transformation q 1 e ɛ q 1 and q 2 e ɛ q 2. In this case, both symmetries lead to the same conservation law, though this situation could also exist in reverse, where the same symmetry of two different Lagrangians leads to different conservation laws.[2] 4.4 Other Conservation Laws Noether s theorem has been extended by physicists to tell us about conservation laws of a system. However, Noether s theorem, when applied in this way, does not work in reverse. A conservation law existing in a system does not imply that a corresponding symmetry exists in the action. Several examples of these exist and are widely used Dynamical Conservations Laws Dynamical conservation laws are conservation laws that correspond to coordinate transformations which do not change the equations of motion, but change constants in the action. These typically involve rescaling variables, which can change the action by an overall multiplicative constant.[2] Solitons Another type of conservation law is associated not with symmetries of the action, but with the topology of the field being examined. One area where 8

9 such conservation laws arise are solitons. Solitons are localized and selfmaintaining solutions to a field equation. Their exact definition is not important for the purposes of this paper, what is important is that every soliton solution to a field equation brings with it a conservation law. [10] The conservation laws which arise from soliton solutions of the field equations have no corresponding symmetry. They are a result of the shape of the field, and interactions between nonlinear elements of the environment. They are called topological conservation laws or topological currents, to differentiate them from Noether currents. [10] 5 Conclusion Noether s theorem is a powerful tool in understanding dynamical systems, but, its applications are limited more than is commonly understood. In spite of that, and despite the numerous restrictions mentioned here, Noether s theorem and its gauge-theoretical neighbor give us powerful methods to anaylyze physical systems without the need to solve the equations of motion. References [1] K. Brading and H. R. Brown. Noether s Theorems and Gauge Symmetries. ArXiv High Energy Physics - Theory e-prints, September [2] Harvey R. Brown * and Peter Holland. Dynamical versus variational symmetries: understanding noether s first theorem. Molecular Physics, 102(11-12): , [3] Harvey R Brown and Peter Holland. Simple applications of noether s first theorem in quantum mechanics and electromagnetism, February [4] N. Byers. E. noether s discovery of the deep connection between symmetries and conservation laws. ArXiv Physics e-prints, jul [5] N.A. LEMOS. Symmetries, noether s theorem and inequivalent lagrangians applied to nonconservative systems. [6] E. Noether. Invariant variation problems. Transport Theory and Statistical Physics, 1: ,

10 [7] Todd. Rowland. lie group. from mathworld a wolfram web resource, created by eric w. [8] Tony Smith. What is a lie group. [9] Eric W. Weisstein. Symmetry group. from mathworld a wolfram web resource. [10] C. V. Westenholz. Topological and noether-conservation laws. Physique Theorique, XXX: ,