Noether charge in multisymplectic geometry

  1. Hi,

    I'm currently looking for the mathematical foundation behind the
    claim, often found in field theory/string theory books that the
    noether charge associated to a symmetry of the lagrangian is the
    generator of that symmetry, ie. its poisson bracket with a field from
    the lagrangian, generates the change in the field.

    Recently I came across the article 'Covariant Momentum maps & field
    theory part I/II' by Gotay and Marsden. In there I've found that the
    concept of a Noether current is in fact given by the covariant
    momentum map in the multisymplectic classical field theory
    formulation. In particular, this map satisfies the mathematical analog
    of noether's theorem as explained in the article.

    In part II of the article they define an instantaneous momentum map,
    loosely defined by the integral of the covariant momentum map over a
    cauchy hypersurface. My question is now, whether this is the
    mathematical object that corresponds to the Noether charge (which in
    physics is defined as the integral of the time component of the
    current over a spacelike hypersurface)? Moreover, if that is so,
    somehow I still can't find an explicit demonstration (in the above
    article or any other mathematical article) that this charge then
    generates the transformation, ie is somehow related to the generators
    of the Lie algebra of the Lie group that encodes the symmetry.

    Could anyone help me out on clarifying these matters?

    Thanks in advance,

    Ygor
     
  2. jcsd
  3. The Noether charge is not the integral of the time-like component of a
    density. It is usually presented that way because you're in flat-space
    and the Cauchy surfaces normally used there are orthogonal with
    respect to a fixed time-like direction.

    On the contrary, it's the integral of a *3-form* density. If Q is the
    charge, then the current density would be J^m, putting J in script to
    emphasize that it's a tensor density. It's integrated against the 3-
    form basis (d^3x)_{m}. For surfaces normal to the timelike direction
    future-oriented, the 3-form corresponding to the surface would be (dx
    ^ dy ^ dz) = (d^3x)_0, in Minkowski coordinates. More generally, you'd
    have
    (d^3x)_{m} = 1/6 epsilon_{mnrs} (dx^n ^ dx^r ^ dx^s).

    Moreover, the charge is not a functional of Cauchy surfaces. It's
    merely presented that way, since the primary context of the concept is
    the Noether theorem and all matters related to it. It's a functional
    of 3-submanifolds (and more generally, 3-chains). So, if S is a 3-
    manifold (or 3-chain) then the charge associated with Q(S) would be
    the integral of the 3-form
    J = J^m (d^3x)_m
    over S.

    The *actual* content of the Noether theorem merely states that Q(S) =
    Q(S') for any two 3-surfaces that comprise the 2 parts of the boundary
    of a 4-region; dO = S - S'. They can even be compact. In fact, I think
    that's the only place where it's truly rigorous. If you were to take O
    to be the non-compact region wedged between two non-compact 3-surfaces
    S and S', then you'd have to add in various technical assumptions
    about how everything behaves in the neighborhood of infinity.
     
  4. On Jun 7, 3:49 pm, ygor.geu...@gmail.com wrote:
    > Hi,
    >
    > I'm currently looking for the mathematical foundation behind the
    > claim, often found in field theory/string theory books that the
    > noether charge associated to a symmetry of the lagrangian is the
    > generator of that symmetry, ie. its poisson bracket with a field from
    > the lagrangian, generates the change in the field.


    This theorem goes back to Hamiltonian mechanics. It generalizes to
    field theory in a straightforward way. Unfortunately, I haven't found
    an explicit demonstration of it in any of my classical mechanics
    books.

    The proof is not particularly complicated, so here it is.

    Consider a Lagrangian function L(v,x) defined on a configuration space
    with coordinates x^i. Let X^i be the coordinate components of the
    vector field that generates a symmetry of the Lagrangian. Then, there
    exists a conserved quantity I(v,x) = L,i X^i. The notation L,ij,klm
    will represent derivatives, with this example representing derivatives
    of L with respect to v^i, v^j, x^k, x^l and x^m.

    The velocity varies under the symmetry transformation as (' is time
    derivative):

    (x^i + eps X^i + ...)' = v^i + eps X^i,k v^k + ... .

    Invariance of the Lagrangian dictates

    L,,i X^i + L,i X^i,k v^k = 0 . (*)

    The Euler-Lagrange equations are

    (L,i)' = L,,i .

    It is easy to show that I(v,x) is a constant of motion:

    I' = (L,i)' X^i + L,i X^i,k v^k (**)
    = L,,i X^i + L,i X^i,k v^k
    = 0 by (*) .

    That's the first half of the theorem you wanted. The other half of the
    theorem states that the conserved quantity I, as a function on the
    phase space, will generate the phase space extension of the
    configuration space vector field X^i.

    The phase space is coordinatized by x^i and p_j = L,j. In terms of the
    the x^i, v^j coordinates, the phase space extension of X^i is

    X = X^i @x^i + X^i,k v^k @v^i ,

    where @x^i and @v^i are the coordinate basis for vector fields. Using
    the chain rule, we express this basis in terms of the @x^i, @p_j
    basis:

    [ @v^i ] [ L,ji 0 ] [ @p_j ]
    [ ] = [ ] [ ] .
    [ @x^i ] [ L,j,i 1 ] [ @x^i ]

    Therefore

    X = X^i ( L,j,i @p_j + @x^i) + X^i,k v^k L,ji @p_j
    = X^i @x^i + ( L,j,i X^i + L,ji X^i,k v^k ) @p_j
    = X^i @x^i + [ (L,,i X^i v^k + L,i X^i,k v^k),j - L,i X^i,j ] @p_j
    = X^i @x^i + [ 0 - L,i X^i,j ] @p_j by (**)
    = X^i @x^i - p_i X^i,j @p_j .

    Using the symplectic form, we can transform this vector field into a
    1-form:

    w = X^i dp_i - p_i X^i,j (-dx^j)
    = X^i dp_i + d(X^i) p_i
    = d(p_i X^i)
    = d(L,i X^i)
    = dI .

    In other words, the above demonstrates that X is precisely the
    Hamiltonian vector field generated by the phase space function
    I(p,x) = p_i X^i, which is presicely the same quantity as the
    conserved
    quantity I(v,x) obtained in the first half of the calculation.

    Hope this helps.

    Igor
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook