Noether's Theorem

Website Links For Theorem

Information About Noether's Theorem

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME
	MORE SHOPPING...

Noether's theorem, published in 1918, holds for all Physical Law s based upon the Action Principle . It is named after the early 20th Century mathematician Emmy Noether . Noether's theorem is a relationship of Classical Mechanics between pairs of Conjugate Variables —if the action is invariant under a shift in one of the two physical variables, then the equations of motion resulting from holding that action stationary conserve the value of the other of the pair of variables. These conjugate pairs also play a crucial role in Quantum Theory —they are the pairs of variables that are related by the Heisenberg Uncertainty Principle (such as Position and Momentum , Time and Energy , Angle and Angular Momentum , etc). MATHEMATICAL STATEMENT OF THE THEOREM Informally, Noether's theorem can be stated as (technical fine points aside): To every differentiable Symmetry generated by local actions, there corresponds a Conserved Current Explanation The word "symmetry" in the above statement refers more precisely to the Covariance of the form that a physical law takes with respect to a one-dimensional Lie Group of transformations satisfying certain technical criteria. The Conservation Law of a Physical Quantity is usually expressed as a Continuity Equation . The formal proof of the theorem uses only the condition of invariance to derive an expression for a current associated with a conserved physical quantity. The conserved quantity is called the ''Noether charge'' and the flow carrying that 'charge' is called the ''Noether current''. The Noether current is defined up to a Divergence less vector field. APPLICATIONS Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example: the invariance of physical systems with respect to spatial Translation (in other words, that the laws of physics do not vary with locations in space) gives the law of conservation of Linear Momentum ; invariance with respect to Rotation gives the law of conservation of Angular Momentum ; invariance with respect to Time translation gives the well known Law Of Conservation Of Energy In Quantum Field Theory , the analog to Noether's theorem, the Ward-Takahashi Identities , yields further conservation laws, such as the conservation of Electric Charge from the invariance with respect to a change in the Phase Factor of the Complex field of the charged particle and the associated the Gauge of the Electric Potential and Vector Potential . The Noether charge is also used in calculating the Entropy of Stationary Black Hole s Calculating the entropy of stationary black holes . PROOF Suppose we have an n-dimensional Manifold , M and a target manifold T. Let $\backslash mathcal\{C\}$ be the Configuration Space of Smooth Function s from M to T. (More generally, we can have smooth sections of a Fiber Bundle over M) Examples of this "M" in physics include: In Classical Mechanics , in the Hamiltonian formulation, M is the one-dimensional manifold R, representing time and the target space is the Cotangent Bundle of Space of generalized positions. In Field Theory , M is the Spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m Real -valued Scalar Field s, φ1,...,φm, then the target manifold is Rm. If the field is a real vector field, then the target manifold is Isomorphic to R3. Now suppose there is a Functional :$\backslash mathcal\{S\}:\backslash mathcal\{C\}\; ightarrow\; \backslash mathbb\{R\},$ called the Action . (Note that it takes values into $\backslash mathbb\{R\}$, rather than $\backslash mathbb\{C\}$; this is for physical reasons, and doesn't really matter for this proof.) To get to the usual version of Noether's theorem, we need additional restrictions on the Action . We assume $\backslash mathcal\{S\}${Link without Title} is the Integral over M of a function :$\backslash mathcal\{L\}(\backslash phi,\backslash partial\_\backslash mu\backslash phi,x)$ called the Lagrangian , depending on φ, its Derivative and the position. In other words, for φ in $\backslash mathcal\{C\}$ :$\backslash mathcal\{S\}\backslash mathrm\{d\}^nx\; \backslash mathcal\{L\}[\backslash phi(x),\backslash partial\_\backslash mu\backslash phi(x),x.$ Suppose we are given Boundary Condition s, ie., a specification of the value of φ at the Boundary if M is Compact , or some limit on φ as x approaches ∞. Then the Subspace of $\backslash mathcal\{C\}$ consisting of functions φ such that all Functional Derivative s of $\backslash mathcal\{S\}$ at φ are zero, that is: :{Link without Title} }{\delta \phi(x)}=0 and that φ satisfies the given boundary conditions, is the subspace of On Shell solutions. (See Principle Of Stationary Action ) Now, suppose we have an Infinitesimal Transformation on $\backslash mathcal\{C\}$, generated by a Functional Derivation , Q such that :$Q\backslash left\backslash mathrm\{d\}^nx\backslash mathcal\{L\}\; ight=\backslash int\_\{\backslash partial\; N\}\backslash mathrm\{d\}s\_\backslash mu\; f^\backslash mu[\backslash phi(x),\backslash partial\backslash phi,\backslash partial\backslash partial\backslash phi,...]$ for all Compact submanifolds N or in other words, :$Q${Link without Title} =\partial_\mu f^\mu(x) for all ''x'', where we set $\backslash mathcal\{L\}(x)=\backslash mathcal\{L\}\backslash partial\_\backslash mu\; \backslash phi(x),x$. If this holds On Shell and Off Shell , we say ''Q'' generates an off-shell symmetry. If this only holds On Shell , we say ''Q'' generates an on-shell symmetry. Then, we say ''Q'' is a generator of a One Parameter Symmetry Lie Group . Now, for any ''N'', because of the Euler-Lagrange theorem, On Shell (and only on-shell), we have \partial_\murac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)} ight]Q {Link without Title} + \int_{\partial N}\mathrm{d}s_\murac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q {Link without Title} But this is the Continuity Equation for the current $J^\backslash mu\backslash ,\backslash !$ defined by :$$ J^\mu\,=\,rac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q {Link without Title} -f^\mu, which is called the Noether current associated with the Symmetry . The continuity equation tells us that if we Integrate this current over a Space-like slice, we get a Conserved quantity called the Noether Charge (provided, of course, if M is Noncompact , the currents fall off sufficiently fast at infinity). Comments Noether's theorem is really a reflection of the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that :$\backslash int\_\{\backslash partial\; N\}\; J^\backslash mu\; \backslash mathrm\{d\}s\_\backslash mu\; =\; 0.$ Noether's theorem is an On Shell theorem. The quantum analog of Noether's theorem are the Ward-Takahashi Identities . Suppose say we have two symmetry derivations Q1 and Q2. Then, {Link without Title} is also a symmetry derivation. Let's see this explicitly. Let's say :$Q\_1${Link without Title} =\partial_\mu f_1^\mu and :$Q\_2${Link without Title} =\partial_\mu f_2^\mu (it doesn't matter if this holds Off Shell or only On Shell ). Then, :$${Link without Title} {Link without Title} =Q_1[Q_2 {Link without Title} ]-Q_2[Q_1 {Link without Title} ]=\partial_\mu f_{12}^\mu where f12=Q1 {Link without Title} -Q2 {Link without Title} . So, :{Link without Title} ]-Q_2 {Link without Title} ])-f_{12}^\mu. This shows we can (trivially) extend Noether's theorem to larger Lie algebras. Generalisation of the proof This applies to ''any'' derivation Q, not just symmetry derivations and also to more general functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields and nonlocal actions. Let ε be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. ε is a Test Function . Then, because of the variational principle (which does ''not'' apply to the boundary, by the way), the derivation distribution q generated by q satisfies q[ε for any ε On Shell , or more compactly, q(x)[S for all x not on the boundary (but remember that q(x) is a shorthand for a derivation ''distribution'', not a derivation parametrized by x in general). This is the generalization of Noether's theorem. To see how the generalization related to the version given above, assume that the action is the spacetime integral of a Lagrangian which only depends on φ and its first derivatives. Also, assume :$Q${Link without Title} =\partial_\mu f^\mu (either off-shell or only on-shell is fine). Then, :$q\backslash mathrm\{d\}^dx\; q[\backslash epsilon[\backslash mathcal\{L\}]$ ::: ::: ::: for all ε. More generally, if the Lagrangian depends on higher derivatives, then : u \phi)} ight]\partial_ u Q {Link without Title} +\partial_ u\left[\left[rac{\partial}{\partial (\partial_\mu \partial_ u \phi)}\mathcal{L} ight] Q[\phi] ight]-\,\cdots ight]=0. EXAMPLES Example 1: Conservation of energy Looking at the specific case of a Newtonian particle of mass ''m'', coordinate ''x'', moving under the influence of a potential ''V'', coordinatized by time ''t''. The Action , ''S'', is: Consider the generator of time translations $Q\; =\; \backslash partial/\backslash partial\; t$. In other words, $Q${Link without Title} =\dot{x}(t). (Quantum field theoreticians would often put a factor of ''i'' on the right hand side.) Note that ''x'' has an explicit dependence on time, whilst ''V'' does not; consequently: : so we can set : Then, The right hand side is the energy and Noether's theorem states that $\backslash dot\{j\}=0$ (i.e. the principle of conservation of energy is a consequence of invariance under time translations). More generally, if the Lagrangian does not depend explicitly on time, the quantity : (called the Hamiltonian ) is conserved. Example 2: Conservation of center of momentum Still considering 1-dimensional time, let

:

Then,

:{Link without Title} -ec{f}

::
::

where is the total momentum, ''M'' is the total mass and is the center of mass. Noether's theorem states:

:.


Example 3: Conformal transformation


Both examples 1 and 2 are over a 1-dimensional manifold (time). For an example involving spacetime, let's work out the case of a Conformal Transformation of a massless real scalar field with a quartic potential in (3 + 1)- Minkowski Spacetime .

For ''Q'', let's consider the generator of a spacetime rescaling. In other words,

:$Q${Link without Title} =x^\mu\partial_\mu \phi(x)+\phi(x).

The second term on the right hand side is due to the "conformal weight" of φ. Note that

:$Q${Link without Title} =\partial^\mu\phi\left(\partial_\mu\phi+x^
u\partial_\mu\partial_
u\phi+\partial_\mu\phi ight)-4\lambda\phi^3\left(x^\mu\partial_\mu\phi+\phi ight).

This has the form of

:
u\phi\partial_
u\phi-\lambda x^\mu\phi^4 ight]=\partial_\mu\left(x^\mu\mathcal{L} ight)

(where we have performed a change of dummy indices) so we can set

:$f^\backslash mu=x^\backslash mu\backslash mathcal\{L\}.\backslash ,$

Then,

:
(\partial_\mu\phi)}\mathcal{L} ight]Q {Link without Title} -f^\mu
:$=\backslash partial^\backslash mu\backslash phi\backslash left(x^$
u\partial_
u\phi+\phi ight)-x^\mu\left(rac{1}{2}\partial^
u\phi\partial_
u\phi-\lambda\phi^4 ight).

Noether's theorem states that $\backslash partial\_\backslash mu\; j^\backslash mu=0$ (as one may explicitly check by substituting the Euler-Lagrange equations into the left hand side).

(Aside: If you try to find the Ward-Takahashi analog of this equation, you'd run into a problem because of Anomalies .)


SEE ALSO

• Charge (physics)

REFERENCES

• English translation of Noether's paper


• Article on Noether's theorem by John Baez


• E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws by Nina Byers


• Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem