Noether ' s theorem and gauge transformations : Application to the bosonic string and CP

New results on the theory of constrained systems are applied to characterize the generators of Noether’s symmetry transformations. As a byproduct, an algorithm to construct gauge transformations in Hamiltonian formalism is derived. This is illustrated with two relevant examples.


I. INTRODUCTION
It is superfluous to emphasize the relevance of gauge theories in modern physics.In spite of this, many aspects of the classical theory of constrained systems-those which have elbow room for gauge transformations-are not completely developed.The aim of this paper is to clarify the role of the Lagrangian Noether theorem in obtaining the generators of Hamiltonian gauge transformations.This is achieved by applying some results recently obtained concerning the relationship between the Hamiltonian and Lagrangian formalisms.I -3 These new results apply to general constrained systems, with first-and second-class constraints, under the only regularity conditions of Ref. 2.
The paper is organized as follows.In Sec.II we set the notation and summarize some of the results of Refs.1-3; they are used in Sec.III to characterize the Hamiltonian generators of a general symmetry Noether transformation.In Sec.IV the specific case of gauge transformations is considered.Section V is devoted to some relevant applications: the bosonic string and the CP ~ -1 model.
All structures are supposed to be COO.Indices of coordinates will be omitted.

II. PRELIMINARY RESULTS
Here we state some of the results needed in Sec.III.For more details see Refs. 2 and 3. Minor changes of notation have been done.
A configuration space Q and a Lagrangian L are given.
We shall always work with natural coordinates such as (q,v) in T(Q) and (q,p) in T(Q)*.
Then the Euler-Lagrange equations for a curve (q(t),p(t») in T(Q) can be written as aq aqav .) Present address: Department of Physics, Princeton University, Princeton, NJ 08540.
(2.5) has the image Mo C T( Q) *, which is assumed to be a submanifold (locally) defined by the mo primary Hamiltonian constraints,p~ (l<ll<m o ) ' The vertical vector fields , where ( a,pO) YJt: =FL * a; (2.7) are a basis for the null vectors of W.
An outstanding object in our development is the operator K 2, which is now understood 4 as a vector field along FL, that is to say, it is a mapping that makes the following diagram commutative: aq aq ap In fact, we shall need K in the time-dependent case, so that we shall add a I at to it: K(q,v,t) =v~+ aL ~+~.
( where SI C T( Q) is the submanifold defined by the primary Lagrangian constraints x~=aYf.l=K•l,b~. (2.21) Bearing all these relations in mind one can prove that where we have introduced m o vector fields along FL: a,if.l a Yf.l(q,u) = ----.

III. CHARACTERIZATION OF NOETHER TRANSFORMATIONS
In the following it will be useful to enlarge our space with a third set of independent coordinates, the accelerations a; that is to say, we shall work in the second tangent bundle We shall consider the operator [which maps functions in dt aq au at Then the Euler-Lagrange equations can be written as where we have defined 3) aq dt au Noether's theorems yield a sufficient condition for a oq(q,u,t) to be a dynamical symmetry transformation (DST) of L, that is to say, to map solutions into solutions.This condition can be written as 8 -1O (3.4) for certain G( q,v,t).We call such a oq a Noether transformation.The acceleration appears linearly in (3.4), so that it splits into two relations 10-12:   aG aG aoq+v-+-=O, aq at aG --Woq=o.au Therefore, there exists G h (q,p,t) (up to primary constraints) such that (3.7) au ap If G corresponds to a Noether transformation, (3.5) and (3.6) set the last two terms to zero.Moreover, assume oq(q,v,t) to be FL projectable.There is Oqh (q,p,t) (up to primary constraints) such that Moreover, Thus there are functions h iL(q,p,t) such that (3.10) that we can assume G h and Dqh chosen in order that Therefore, we conclude from (3.9) and (3.11) that (3.8) becomes (3.12) Conversely, suppose we have G h (q,p,t) satisfying relation (3.12) and define Dqh ' Dq,and G as in (3.11), (3.9), and (3.7).Then we have aG lav = W FL * (aGhlap) = W Dq, which is (3.6), and the identity for K'G h [(3.8)] shows that (3.5) also holds; that is to say, (3.4) is satisfied.We have proven the following theorem.
Theorem 1: An infinitesimal projectable function Dq(q,V,t) is a Noether transformation if there exists G h (q,p,t) such that K-G h = 0 and 8q = FL *{q, G h }. .Now we make use of this Lagrangian result to derive a sufficient condition for a G h (q,p,t) to generate a Hamiltonian DST in the sense that (3.13 ) Theorem 2: An infinitesimal function G h (q,p,t) satisfying K• G h = 0 generates a Hamiltonian DST.
We call such a DST a Hamiltonian Noether transformation.We have shown that Dq: = FL *{q,G h } isa Lagrangian DST.Taking into account the equivalence of both formalisms,z we only need show D(aL lav) = FL *{p,G h }.To this end we write the following identity, which can be obtained using (2.9) and the chain'  Finally, we want to express (3.12) in an equivalent way, which will prove to be useful in the case of gauge transformations.Application of (2.14) to (3.12) shows that FL *{Gh'rp~} = 0, that is to say, {Gh'rp~} = 0. (3.16) Now (2.13) leads to FL *({Gh,H} + aGhlat) = 0, which implies ( 3.17) Conversely, by (2.13), (3.16) and (3.17) imply (3.12).Therefore, the following theorem holds.
It can be shown that these sufficient conditions [( 3.16) and (3.17)] are in fact very close to those that are necessary.13Notice, also, from (3.17) that G h is a constant of motion.Moreover, in a constrained system G h is a first class function because it must be tangent to the final constraint manifold.

IV. HAMILTONIAN GAUGE TRANSFORMATIONS
The preceding results apply to DST in general dynamical systems.Now we consider the specific case of gauge transformations, that is to say, DST depending on arbitrary functions and their derivatives.Thus we are necessarily dealing with a constrained system.We will write a generator G (q,p,t) of a gauge transformation in the form  4.4) can be seen as a mechanism to construct a gauge transformation.Since G is first class, the G k are also first class.To be precise, the G k are first-class constraints: Let us prove this inductively; it is obvious for G o [ (4.3)].Suppose we have chosen H to be first class (which is always possible; for instance, the H (1+ I) reached in Ref. 2).Then if G k is a first-class constraint, {Gk,H} is as well.Therefore, (4.4) implies that G k + I is also a first-class constraint.Notice, also, that G k + I + {Gk,H} is a primary first-class constraint.
The algorithm can be applied in the following way (see, also, Ref. 14, which proposes an algorithm to construct the gauge generator when no second class constraints are present): His a first-class Hamiltonian and Go = primary first-class constraint, (4.5) Gk+ 1 = -{Gk,H} + primary first-class constraints. (4.6) One must play with this indeterminacy in order to let the test (4.2) hold.It is worth observing that the simpler form of a primary first-class constraint may not be suitable to begin (4.5).
There is no guarantee that this algorithm has a solution; however, it is reached in usual computations.Moreover, in these cases one can choose G k = ° for k>f + 1 (if the stabilization algorithm ends at thefth step).For this reason the generator is usually written as no tertiary constraints appear and we are left with five (TI oo , TI oI , TI II' H, and T) first-class constraints.
We have three primary first-class constraints, so we expect three independent gauge transformations.The algorithm for constructing a canonical gauge generator starts by selecting a combination of primary first-class constraints.In order to simplify the expressions and taking into account that the three primary constraints give only two secondary constraints, let us consider the following combinations: ( 5.8) Thus we see that the generator starting with ({J w has only one piece: G w = f du Ew(U) (gooTIoo + gOITI OI + gIlTI Il )• (5.9) '(a({Jw(d) + {3 ((JI (d) + r({J2 (u') j, where a = a(u,d), etc.Then G I (u) = -H(u) + f du '(aq.>w(d)

Now let us consider ({JI and apply the algorithm
+ primary first-class constraints. We need to compute (5.24 ) which gives the correct gauge transformations.In this condition (4.2) is trivially satisfied in a natural way. --v--.
Therefore, generators of Hamiltonian Noether transformations close under the Poisson bracket.
where € is an arbitrary function of time and €( -k(t) is a primitive of order k.As a result of the arbitrariness of €, conditions (3.16) and (3.17) split into Go=O,