aa r X i v : . [ g r- q c ] O c t Lectures on Constrained Systems
Ghanashyam Date
The Institute of Mathematical SciencesC.I.T. Campus, Tharamani, Chennai, 600 113, India. email: [email protected] reface
These lecture notes have been prepared as a basic introduction to the theory of constrainedsystems which is how basic forces of nature appear in their Hamiltonian formulation. Onlya basic familiarity of Lagrangian and Hamiltonian formulation of mechanics is assumed.The First chapter makes some introductory remarks indicating the context in which varioustypes of constrained systems arise. It distinguishes constrained systems for which the equa-tions of motion are uniquely specified from those for which the equations of motion are not uniquely determined. The focus of the lectures is on the latter types of constraints. Thenotations that will be used is introduced here.In the second chapter, the general features are introduced in the familiar example of source-free electrodynamics.In the third chapter, features seen in electrodynamics are abstracted in the simpler contextof systems with finitely many degrees of freedom. How constraints arise in a Lagrangian andin a Hamiltonian formulation is discussed.In the fourth chapter, the Dirac’s algorithm for discovering constraints and their classificationis discussed in a sufficiently general context.In the fifth chapter, a special class of constrained systems for which the Hamiltonian itselfis a constraint is discussed. Such systems arise in the context of general relativity and leadto a host of issues of interpretation of ‘dynamics’.A set of exercises is also included for practice in the last chapter.These set of eight lectures were given at the
Refresher Course for College Teachers held atIMSc during May-June, 2005.June 4, 2005 Ghanashyam Date2 ontents hapter 1Introductory Remarks Let us begin by recalling some elementary understanding of what one means by mechanics.There are two distinct parts: (a) kinematics and (b) dynamics. Kinematics specifies the(possible) states of a given system while dynamics specifies how a system evolves from onestate to another state. The central problem one wants to solve is to find the state of systemat some time t given its state at an (earlier) time t . While there are various types of ‘statespaces’ possible, we will be dealing with those systems for which the state space is a manifold (finite or infinite dimensional) and an evolution which is of the continuous variety.There are three distinct tasks before us (i) devise a framework or a calculational prescriptionwhich will take as input a state space, some quantity reflecting/encoding a dynamics andgive us a corresponding rule to evolve any given state of the system. A familiar exampleis the Lagrangian framework wherein the state space is described in terms of positions andvelocities, dynamics is encoded in terms of a Lagrangian function and the variational principleleads to the Euler-Lagrange equations of motion. (ii) Use physical, qualitative analysis of agiven physical system to associate a particular state space and a particular Lagrangian (say)with the physical system and (iii) devise methods to obtain generic evolutions in sufficientlyexplicit terms.The last task is where one will make ‘predictions’ and is the most relevant in applications (egengineering). Here the issues such as sensitivity to initial conditions etc play an importantrole. This task can be addressed only after the first two are specified. This is not the aspectwe will discuss in these lectures. Some of it will be discussed in the NLDL part of the course.To be definite, let us consider the Lagrangian framework. Thus we will have some config-uration space , Q , which is some n dimensional manifold with coordinates q i , i = 1 , , · · · n .The dynamics is encoded in a function L ( q i , ˙ q i , t ) and the equations of motion are the Euler-Lagrange equations of motion, δS = δ Z dtL ( q, ˙( q )) = 0 ⇒ ddt ∂L∂ ˙ q i − ∂L∂q i = 0 , ∀ i. (1.1)Questions: How general can Q be? How general can L be? Can we always obtain equationsof motion via the variational principle?Generally, one takes Q to be an n-dimensional manifold. However, this need not be R N ,although usually it is. For instance one may begin with n particles moving in 3 dimensions,4o that Q is 3 n dimensional, but there could be restrictions such as distance between any twoparticles is always fixed (rigid body) or that particles must have velocities tangential to thesurface of a ball. In the latter case, the relevant Q would be the 2-sphere which is a compactmanifold without boundary. In both cases the dimension of the relevant configuration spaceis smaller that what we began with. We could also imagine confining the particles to a 3dimensional box in which case the relevant configuration space is a bounded portion of R n .In short, the possible motion of the system may be constrained . A majority of applied me-chanics problems have to deal with such constrained systems and there is a massive recenttreatise on such systems [1]. Typically, such constrained systems are described by specifyingrelations f α ( t, ~r, ~v ) = 0. These relations are required to be functionally independent , mu-tually consistent, and valid for all possible forces . Some terminology that you might havecome across classifies the constraints as: (1) Relations independent of velocities are termed positional/holonomic/finite constraints ; (2) Velocity dependent ones are called velocity con-straints which are further divided into scleronomic (time independent), rheonomic (timedependent), holonomic (integrable), non-holonomic (non-integrable). Constrained systemsof this variety, serve to restrict the configuration space and also possible motions, but the Euler-Lagrange equations are always solvable for accelerations, so that dynamics is uniquelyspecified.
We will not be dealing with this type of constrained systems.The constrained systems that we will deal with arise in situations where the well moti-vated choices of the configuration space and the Lagrangian, do not specify the equationsof motion uniquely. In the Lagrangian framework, these correspond to the so called singu-lar
Lagrangians. Analysis of such systems is better carried in the Hamiltonian frameworkwherein one arrives at an understanding of gauge theories and we will be essentially focusingon such constrained Hamiltonian systems .From the point of view of usual applications of classical mechanics, such systems wouldappear quite exotic and possibly ‘irrelevant’. However all the four fundamental interactionsthat we know of, when cast in a Lagrangian or Hamiltonian framework, precisely correspondto the kind of constrained systems we will discuss. An understanding of these basic forces atthe classical structural level is crucial also for constructing/understanding the correspondingquantum theories. In the context of general relativity, the constrained nature of the theorythrows up challenging conceptual and interpretational issues. With these reasons as primarymotivations, we will discuss constrained systems.We will also use certain notations which are introduced below.1. I will freely use terms such as ‘manifold’, ‘tensors’ etc. Here is a very rapid, heuristicintroduction to these terms.The idea of a manifold (actually a differentiable manifold), is invented to be able todo differential and integral calculus on arbitrarily complicated spaces such as (say) thesurface of an arbitrarily shaped balloon. The basic definition of differentiation involvestaking differences of values of a function at two nearby points eg., f ( x + h ) − f ( x ),dividing it by h and taking the limit h →
0. This is fine when x, h etc are numbers.But when we go to the surface of a balloon, we do not know how to implement suchdefinitions eg how to take the ‘difference of two points’ on the surface. The way theidea of differentiation is captured is to assign numbers to points and then use themin the usual manner. This assignment can be thought of as pasting small pieces of agraph paper on the surface, and reading off the numbers corresponding to the pointsjust below the numbers. This pasting gives a set of local coordinates in that portion5f the surface. Clearly there is huge freedom in the choice of the pieces of graphpaper as well as in the pasting. Thus, while local coordinates can be introduced, thearbitrariness must be respected. Changing these assignments is called local coordinatetransformations . Likewise, quantities such as tiny arrows stuck on the surface can bedescribed in terms of components of the arrows with respect to the coordinate axesprovided by the graph paper, When the graph paper is changed, these componentschange, but not the arrows themselves. Consequently, the two sets of componentsmust be related to the local coordinate transformations in a specific manner so thatboth the sets refer to the same arrow. The arrows are an example of a vector field which can be thought of as a collection of their components, transforming in a specificmanner. The components are denoted as quantities with arbitrary number of upperand lower indices . These must transform as,( X ′ ) i , ··· ,i m j , ··· ,j n ( y ( x )) = ∂y i ∂x r · · · ∂y i m ∂x r m ∂x s ∂y j · · · ∂x s m ∂y j m ( X ) r , ··· ,r m s , ··· ,s n ( x ) (1.2)Objects represented by such indexed quantities have a meaning independent of thechoice of local coordinates and are called Tensors of contravariant rank m and covari-ant rank n . Equations involving tensors (correctly matched index distribution) are covariant (or form invariant) with respect to local coordinate transformations. This isall that we will need to know.2. In the context of Lagrangian framework, the generalized coordinates will be denotedby q i . These are not necessarily Cartesian and are to be thought of as arbitrary localcoordinates on the configuration space manifold. Consequently, we will occasionallyalso comment on whether various equations/conditions are ‘covariant/invariant’ undercoordinate transformations. For these purposes, tensor notation will be used freely. La-grangian will always be taken to be a function of generalized coordinates and velocitiesand independent of time.3. In the context of Hamiltonian framework, the generalized canonical coordinates willbe denoted as q i , p i . It will be convenient to subsume them as 2 n coordinates ω µ . Thephase space will be denoted as Γ which is a 2 n dimensional manifold. On this there isa distinguished antisymmetric rank 2 tensor, Ω µν ( ω ) which satisfies further propertiesnamely, ∂ λ Ω µν + cyclic = 0 . (1.3)The antisymmetric matrix Ω µν is assumed to be invertible and its inverse is denotedas Ω µν , Ω µλ Ω λν = δ µν .Such an antisymmetric tensor is called a symplectic form . The coordinates in whichthe matrix takes the block off-diagonal formΩ µν = (cid:18) − (cid:19) , Ω µν = (cid:18) −
11 0 (cid:19) (1.4)are the canonical coordinates and these are guaranteed to exist by Darboux theorem.4. The usual Poisson bracket of two functions f ( ω ) , g ( ω ) get expressed as, { f ( ω ) , g ( ω ) } = Ω µν ∂f∂ω µ ∂g∂ω ν (1.5)6he (1.3) condition implies the Jacobi identity of Poisson brackets.For every function f on the phase space, one can associate the so-called Hamilto-nian vector field, v µf := Ω µν ∂ ν f . Such Hamiltonian vector fields generate infinitesimalcanonical transformations as, ω µ → ω ′ µ := ω µ + ǫv µf ( ω ) . (1.6)It is easy to check that the symplectic form is invariant under these coordinate trans-formations (and hence these are called canonical transformations). In particular, theHamiltonian function generates time dependent changes in coordinates which is noth-ing but the dynamical evolution. Hence, dynamical evolution can be viewed as a“continuous unfolding of canonical transformations”. Taking ǫ := δt, f = H impliesthe Hamilton’s equations of motion, dω µ dt = Ω µν ∂H∂ω ν (1.7)Note that these are not just any system of first order, ordinary differential equation,but have a specific form involving the antisymmetric tensor Ω µν because of which H is constant along solutions of equations of motion. Furthermore, one also has theLiouville theorem regarding conservation of phase space volumes.We will begin our discussion by taking the example of source free electrodynamics.7 hapter 2Hamiltonian Formulation ofElectrodynamics We will begin with the usual Maxwell equations, put them in the four dimensional rela-tivistic form, arrive at an action formulation from which we will go to the Lagrangian andHamiltonian form. To be able to use the four dimensional relativistic tensor notation, weneed to choose a set of conventions.Maxwell Equations: ~ ∇ · ~E = ρ , ~ ∇ × ~E + ∂ ~B∂t = 0 (2.1) ~ ∇ · ~B = 0 , ~ ∇ × ~B − ∂ ~E∂t = ~j (2.2) ∂ρ∂t = − ~ ∇ · ~j (2.3)Conventions: • Write all quantities without derivatives and with arrows as quantities with upper in-dices i.e. as contravariant tensor , eg, ~a ↔ ( a , a , a ) ↔ a i and vector derivatives asderivatives with lower indices ( covariant tensor ), eg, ~ ∇ ↔ ( ∂ , ∂ , ∂ ) ↔ ∂ i .Thus, electric field, magnetic field and vector potential are contravariant tensors whilethe gradient operator is a covariant quantity. The cross product explicitly gives acontravariant quantity. • Quantities with upper indices are related to those with lower indices by, a i = − a i , a = a i.e. a µ := η µν a ν with η µν = diag(1 , − , − , −
1) = η µν . • Cross products are expressed as, ( ~ ∇ × ~a ) i := ǫ ij k ∂ j a k , where , ǫ = 1 = − ǫ . It thenfollows that, ǫ ij k ǫ kmn = − ( δ im δ jn − δ in δ jm ) . (2.4)Maxwell equations now become, ∂ i E i = ρ , ǫ ij k ∂ j B k − ∂E i ∂t = j i (2.5)8 i B i = 0 , ǫ ij k ∂ j E k + ∂B i ∂t = 0 (2.6)Define, j := ρ, F i := − E i , F ij := − ǫ ij k B k . It is easy to check that(2.5) → ∂ ν F νµ = j µ , (2.7)(2.6) → ∂ µ F νλ + cyclic = 0 = ∂ µ F νλ + cyclic (2.8)It follows that B i = ǫ i jk F jk and the identification of the electric field matches with theusual definition with A µ ↔ ( A := Φ , ~A ).This allows us to think of Maxwell equations as tensor equations involving 4 dimensionaltensors. If one transforms the coordinates x µ as x ′ µ := Λ µν x ν and transforms the F, j quanti-ties by tensor rules (index-by-index action), then evidently the Maxwell equations are forminvariant or covariant for all invertible matrices Λ µν . Remember though that we have alsoused specific rules to relate the upper and the lower indexed quantities. These rules mustalso be respected by the primed observer (coordinates) i.e. η µν must also be invariant underthe coordinate transformations and this restricts the Λ µν to be the familiar Lorentz transfor-mation matrices. We thus also see the Lorentz covariance of Maxwell equations. Incidentally,operating by ∂ µ on (2.8) and using (2.7) one can deduce the wave equation, (cid:3) F µν = 0 andinvariance of (cid:3) := η µν ∂ µ ∂ ν is precisely the requirement of Lorentz invariance.Note that the 4-tensor notation, is just a compact notation to write Maxwell equations andthe physical property of electrodynamics being Lorentz covariant is encoded by the invarianceof (cid:3) which in turn requires invariance of η . The compact notation helps to keep track of theLorentz covariance property.Now we would like to see if these equations can be obtained from an action principle and weall know that the answer is of course yes. For our purposes, it is sufficient to consider thesource free case and we will now take j µ = 0.If F µν are treated as basic variables describing electrodynamics, then we have 8 first orderpartial differential equations for six quantities. Usual equations of Lagrangian frameworkare second order (in t ) equations for configuration space variables. However one notices that(2.8) can be identically solved by putting F µν := ∂ µ A ν − ∂ ν A µ . This is always possible to dolocally. Now the remaining 4 equations become 4 second order equations for the 4 quantities A µ . At least now the number of equations equals the number of unknowns. However, thedefinition of F in terms of A does not determine A uniquely (not even up to a constant); A ′ µ = A µ + ∂ µ Λ gives the same F . Therefore, although we have correct number of equations,these equations do not suffice to determine the candidate dynamical variables, A , uniquely.Modulo this observation, let us go ahead any way by thinking of A µ ( x α ) as ‘configurationsspace variables’.It is a very easy exercise to check that if we define an action, S [ A ( x )] := Z dt Z d x (cid:18) − F µν F µν (cid:19) := Z d x L ( A µ , ∂ α A β ) , (2.9)then its stationarity condition gives the Maxwell equations expressed in terms of A . This ofcourse is an example of a field system i.e. a dynamical system with infinitely many degreesof freedom , A µ ( t, x i ) with µ, x i serving as labels. To proceed further, let us introduce some9orking rules and notations: derivatives of the Lagrangian density etc will be denoted as δ L ( x ) δA µ ( x ′ ) with the rules, δA µ ( x ) δA ν ( x ′ ) = δ νµ δ ( x i − x ′ i ) , δ∂ ν A µ ( x ) δA λ ( x ′ ) = δ λµ ∂ ν δ ( x i − x ′ i ) . (2.10)Now let us try to get to a Hamiltonian formulation. Our basic configuration space variablesare A µ ( t, x i ) while the Lagrangian is L = R d x ( − / F µν F µν . The conjugate momenta aregiven by, π µ ( x ) := Z d x ′ δ L ( x ) δ∂ A µ ( x ′ ) = − F µ ( x ) . (2.11)Clearly, π = 0 identically while π i = − F i . Therefore, π i = − F i = − ∂ A i + ∂ i A . Thisallows us to express the ‘velocities’ ∂ A i = − π i + ∂ i A which is needed in getting to theHamiltonian form. However, we cannot do so for ∂ A !The canonical Hamiltonian is defined as, H c = Z d x (cid:20) π µ ∂ A µ + 14 F µν F µν (cid:21) (2.12)= Z d x (cid:20) π i ( ∂ A i − ∂ i A + ∂ i A ) − (cid:0) F i (cid:1) + 14 F ij F ij + π ∂ A (cid:21) = Z d x (cid:20) − π i π i + 14 F ij F ij − A ( ∂ i π i ) + π ∂ A (cid:21) (2.13)In the last equation, a partial integration has been done. The first two terms are theusual electromagnetic field energy density, ~E · ~E + ~B · ~B . The last term depends only on π , ∂ A and is not a function of phase space variables, it contains a velocity. If we naivelyconsider the Hamilton’s equation of motion for A , we get an identity, implying that thetime dependence of A is undetermined. The equation of motion for π however leads to ∂ π = ∂ i π i . Observe that if we drop the last term by setting π = 0 at some initialtime, and require this condition to hold for all times, then we will need ∂ i π i = 0 for all timewhich is just one of the Maxwell equations (the Gauss Law). This is also suggested by theLegendre transformation involved in going from the Lagrangian to the Hamiltonian – we arerequired to use the definition of the momenta. Hence, in the Hamiltonian we should nothave the last term. Since π = 0 from definition at all times , we must also obtain ∂ π = 0from the Hamilton’s equation of motion. Thus it is self consistent to interpret the droppingof the last term as imposition of a constraint, π = 0 on the phase space of ( A µ , π µ ) andthe Gauss law following as a consistency condition. Since A remains undetermined, we willreplace A in the third term by an arbitrary function of time, λ ( t ). Since the Gauss law isalso independent of positions/velocities, we will interpret it also as a constraint, χ := ∂ i π i .We will refer to π = 0 as a primary constraint and χ = 0 as a secondary constraint , becauseit is needed for the primary constraint to hold for all times.Let us consider now the canonical transformations generated by the two constraints. Itfollows, δ ǫ A µ ( x ) = (cid:26) A µ ( x ) , Z d x ′ ǫ ( x ′ ) π ( x ′ ) (cid:27) = δ µ ǫ ( x ) (2.14) δ η A µ ( x ) = (cid:26) A µ ( x ) , Z d x ′ η ( x ′ ) ∂ i π i ( x ′ ) (cid:27) = − δ iµ ∂ i η ( x ) (2.15) δπ ( x ) = 0 (2.16)10learly, δ ǫ A ( x ) = ǫ ( x ) , δ η A i ( x ) = − ∂η ( x ). Choosing ǫ ( x ) := − ∂ η ( x ), reveals that thecanonical transformations generated by the constraints are nothing but the usual gaugetransformations .Let us summarize.1. Maxwell equations can be written in a manifestly relativistic form.2. These can be derived from relativistic action principle treating A µ ( t, x i ) as generalizedcoordinates. In the Lagrangian formulation, the ‘matrix of second derivatives’, δ L δ ( ∂ A µ ( x )) δ ( ∂ A ν ( x ′ )) := M µν ( x, x ′ ) = (cid:0) − η µν + η µ η ν (cid:1) δ ( x − x ′ ) , (2.17)is non-invertible .3. In the Hamiltonian formulation, there are constraints . These generate canonical trans-formations which are the familiar gauge transformations .4. The canonical Hamiltonian has the form which contains the constraints with arbitraryfunctions of time as coefficients.5. We also know that while manifest special relativistic formulation requires us to use4 components of the vector potential as basic configuration space variables (i.e. 4degrees of freedom per point), physically there are only 2 degrees of freedom (thetwo polarizations). Thus the constraints inferred above have something to do withidentifying physical degrees of freedom .We will keep these in mind and try to abstract these features in a general framework. Onthe one hand we will simplify by restricting to finitely many degrees of freedom and at thesame time generalize to arbitrarily complicated systems.11 hapter 3Constrained Lagrangian andHamiltonian Systems Consider a dynamical system with finitely many degrees of freedom, described in a La-grangian framework. Let Q be an n -dimensional manifold with local coordinates q i servingas generalized coordinates. Let L ( q, ˙ q ) be the Lagrangian function of the dynamical sys-tem and S [ q ( t )] := R dtL ( q, ˙ q ) being the action. The Variational Principle leads to theEuler-Lagrange equations of motion, ddt (cid:18) ∂L∂ ˙ q i (cid:19) = ∂L∂q i i = 1 , , · · · , n , (3.1) (cid:18) ∂ L∂ ˙ q i ∂ ˙ q j (cid:19) ¨ q j := M ij ( q, ˙ q ) ¨ q j = ∂L∂q i − ∂ L∂ ˙ q i ∂q j ˙ q j := Q i ( q, ˙ q ) . (3.2)Usually, the matrix M ij is invertible which allows us to solve for the accelerations, ¨ q i =( M − ) ij Q j . This implies that given position and velocity at an instance, one can alwaysdetermine the dynamical trajectory at other instances.If on the other hand the matrix M ij is not invertible, then let 1 ≤ r ≤ n , denote itsrank. Then there exist ( n − r ) independent h ia , vectors satisfying M h = 0 and we cannotsolve for all the accelerations uniquely . The general solution for the accelerations will be,¨ q i = ¨ q i + P n − ra =1 α a h ia and the first term is the solution of the inhomogeneous equation.Furthermore, M h = 0 implies h T M = 0 = h ia Q i ( q, ˙ q ) = 0 for a = 1 , · · · ( n − r ). Thus on theone hand the equations of motion are not uniquely specified and on the other hand there are( n − r ) relations among the 2 n positions and velocities. If these relations are not satisfiedidentically, then in particular, they correspond to restrictions on the initial data, q i (0) , ˙ q i (0).As an example, consider a system with a one dimensional configuration space with coordinate q and with a Lagrangian, L = f ( q ) ˙ q − V ( q ). Clearly, M = 0 and Q = − V ′ ( q ). Theacceleration is given by ¨ q = αh and hQ = 0 ⇒ V ′ ( q ) = 0. If V = 0 then the equation issatisfied identically while for non-zero V , the position must be at an extremum of V (whichmay not exist!).Let us see what implications are there for a corresponding Hamiltonian framework. Firstobservation is that for a singular Lagrangian, the definition of conjugate momentum p i := ∂L∂ ˙ q i cannot be inverted to solve for the velocities in terms of the momenta. This is essentially the12 nverse function theorem namely, y i = f i ( x ) can be inverted to get x i = g i ( y ) provided ∂f i ∂x j isan invertible matrix. (Strictly speaking, the n momenta are n functions of the 2 n positionsand velocities and hence one should be invoking the implicit function theorem , such fineprints are ignored here.)Let us define the Hamiltonian by H := p i ˙ q i − L ( q, ˙ q ) and consider its variation, δH = ˙ q i δp i + (cid:18) p i − ∂L∂ ˙ q i (cid:19) δ ˙ q i − ∂L∂q i δq i (3.3)If we treat the q, ˙ q, p all as independent variables, then the ‘Hamiltonian’ is clearly a functionof all of these. Let us use the definition of momenta (i.e. restrict to a 2 n dimensional sub-manifold of the 3 n dimensional space). On this sub-manifold, the middle term vanishes andthe Hamiltonian becomes a function only of positions and momenta in the sense that it variesonly when q i , p i are varied. This is independent of whether velocities can be solved for interms of momenta and positions. The next step in the usual case of non-singular Lagrangiansis when one infers the Hamilton’s equations of motion as,˙ q i = ∂H∂p i , ˙ p i = − ∂H∂q i . (3.4)This assumes that (a) the variations δp i , δq i are independent and (b) Hamiltonian is a function only of p i , q i . Both these statements fail for singular Lagrangians.To see this, notice that variation of the Hamiltonian being given in terms of variations ofmomenta and positions, depends on evaluating the variation of the Hamiltonian on the sub-manifold defined by p i = ∂L∂ ˙ q i . If the variations are also to respect this condition, we musthave δp i = M ij δ ˙ q j + ∂ L∂ ˙ q i ∂q j δq j (3.5) h ia δp i = 0 + h ia ∂ L∂ ˙ q i ∂q j δq j (3.6)which immediately shows that the variations δp i , δq i are not independent . Consequently, onecannot deduce the Hamilton’s equations of motion. We have already noted that for singularLagrangians, the velocities cannot be eliminated in favour of momenta and positions. Thevariations being not independent means that there are relations among the phase spacevariables p i , q i , namely, φ a ( q, p ) = 0.Let us summarize.A Lagrangian system is said to be singular if the matrix M ij of second derivatives of theLagrangian with respect to the velocities, is singular. This has the consequences that (a) theaccelerations are not determined uniquely i.e. contain arbitrary functions of time and (b)it could imply relations among velocities and positions which may not even be consistent.It is possible to isolate the accelerations which are determined uniquely. However we willcarry out such analysis in the context of a Hamiltonian framework. The property of being asingular Lagrangian is independent of any choice of local coordinates. Generically, singularLagrangian systems are also called constrained systems [3].In a Hamiltonian formulation obtained from a Lagrangian, the singular nature of the La-grangian manifests as certain relations among the phase space variables. We will therefore13efine a constrained Hamiltonian System to be a system with a 2 n dimensional phase spacemanifold Γ with local coordinates ( q i , p i ) ↔ ω µ , on which is given a Hamiltonian function, H ( q i , p i ) := H ( ω ) together with a set of relations φ a ( ω ) = 0 , a = 1 , · · · k ( < n ). These rela-tions are referred to as Primary Constraints . By definition, the k constraints are functionallyindependent i.e. the k differentials, dφ a ( ω ) (or the k vectors, ∂ µ φ a ) are linearly independent.The sub-manifold defined by φ a = 0 will be denoted by Σ and is called the constraint surface .It is a 2 n − k dimensional sub-manifold of Γ. Note that a constraint surface is not a phasespace in general, i.e. does not have a symplectic form (eg when k is odd).We will focus entirely on the constrained Hamiltonian systems and analyze various possibil-ities of types of such systems. The aim will be to have a procedure for obtaining equationsof motion in the Hamiltonian form paying attention to the constraints. This means that wewant to have a variational principle for paths in Γ, which will lead to ‘Hamilton’s equationsof motion’ for some suitable Hamiltonian function and such that the possible dynamical tra-jectories (in Γ) either remain confined to the constraint surface, Σ or never intersect it. Ineffect, we can continue to work in the given phase space and use a new
Hamiltonian functionso that dynamics effectively respects the constraints. This is achieved by using the methodof Lagrange multipliers.Introduce k Lagrange multipliers, λ a and define a new Hamiltonian function H := H + λ a φ a ,which matches with the given Hamiltonian H on the constraint surface , Σ. Defining anaction, S [ ω ( t ) , λ ( t )] := Z dt (cid:20) ω µ Ω µν ˙ ω ν − H ( ω ) − λ a ( t ) φ a ( ω ) (cid:21) , (3.7)and invoking its stationarity, δS = 0, leads to the equations of motion, dω µ dt = Ω µν (cid:18) ∂H ∂ω ν + λ a ∂φ a ∂ω ν (cid:19) and φ a ( ω ) = 0 , a = 1 , · · · , k . (3.8)Thus we obtain equations of motion in a Hamiltonian form and also the constraint equations.We have succeeded in having a new Hamiltonian dynamics defined for trajectories in Γ. Wehave now to ensure that the trajectories are such that if an initial point is on the constrainedsurface, then the whole trajectory also remain on the constrained surface. If this propertycan be ensured, then it also follows that no trajectory can enter and leave Σ, since theequations are first order.Observe that if the value of φ a for any given a is preserved under evolution (i.e. ˙ φ a = 0along a trajectory), then the trajectory is confined to the 2 n − φ a =constant. However we only need the trajectory to be confined to Σ, so the value ofeach constraint need not be preserved exactly . Therefore we need not have ˙ φ a = 0 along all trajectories but only along those trajectories which lie in Σ. This is ensured by requiringthat the Poisson bracket of the constraints with the Hamiltonian H be weakly zero . SincePoisson brackets can be evaluated at any point of the phase space, we can evaluate these atpoints on the constraint surface and weak equality/equations refer to Poisson brackets beingevaluated at points of Σ and are denoted by ‘ ≈ ’ [2]. Strong equations/equalities are valid ina neighbourhood of Σ and are denoted by the usual ‘=’. In particular a strongly vanishingfunction vanishes weakly and so do all its partial derivatives . dφ a dt = { φ a , H } + λ b { φ a , φ b } ≈ k conditions ensure that a trajectory beginning on the constrained surface remains onthe constraint surface and our goal is reached provided (3.9) holds on Σ.14here are several possibilities now [2]. If the k × k matrix of Poisson brackets of the constraints is non-vanishing , then the con-sistency conditions can be viewed as a matrix equation at each point on Σ for the Lagrangemultipliers λ a . Since this matrix is antisymmetric, it is non-singular only if when k is even.In this case, all Lagrange multipliers are determined and we do have a Hamiltonian dynam-ics whose trajectories either lie on the constraint surface or never intersect it. Genericallyhowever the matrix is singular. As seen in the context of singular Lagrangian, this meansthat (a) some multipliers are necessarily undetermined and (b) some linear combinations of { φ a , H } must vanish on Σ. Again we have several possibilities. Either (i) { φ a , H } ≈ , ∀ a and all linear combinations are weakly zero, or (ii) the linear combinations vanish weakly provided some further functions vanish in which case we refer to these as secondary con-straints , or (iii) there are no points of Σ at which the linear combinations vanish in whichcase we say that the Hamiltonian system is inconsistent .It could also happen that the matrix is zero on the constraint surface. This could happenfor instance, if { φ a , φ b } = C abc φ c . (Recall the Maxwell example). In such a case, there is noequation for the Lagrange multipliers and all Lagrange multipliers are undetermined. Thiscase can be thought of as a special case of rank of the matrix being zero.If we encounter the inconsistent case, we throw away our formulation of the system andstart all over again. In the case (i), we have reached our goal but have to live with someundetermined Lagrange multipliers (and hence evolution). This will turn out to be the mostinteresting case. In the case (ii), we have to now demand that the Poisson bracket of sec-ondary constraints with the Hamiltonian H must vanish on Σ. Once again we will encountersimilar cases as above and we have to repeat the analysis – we will either satisfy the condi-tions identically with some Lagrange multipliers determined or encounter tertiary constraints or encounter inconsistency. Since the total number of constraints cannot be larger that 2 n (else no initial condition will be left!), the process must terminate. Barring inconsistentsystems, we will eventually end up with some Lagrange multipliers being determined, someundetermined and with possibly additional constraints χ A ≈ , A = 1 , · · · , l .Note: We began by requiring the trajectories to be confined to Σ and found as a consistencyrequirement that the goal cannot be attained for all points of Σ. We need to restrict furtherto a sub-manifold Σ ′ ⊂ Σ, due to the secondary constraints. Thus, the dynamics defined by H on Γ , will correspond to a constrained dynamics relative to Σ ′ defined by all constraintsbeing weakly zero . The dimension of Σ ′ is of course 2 n − k − l . Notice that even if we began byrequiring consistency condition to hold on Σ, extending it to hold on Σ ′ , does not contradictthe previous condition since Σ ′ is a sub-manifold of Σ. The weak/strong equation now referto Σ ′ .To summarize: Beginning with a phase space Γ, a Hamiltonian H and a set of primaryconstraints φ a , we can construct a Hamiltonian dynamics on Γ such that its trajectoriesare either confined to the constraint surface Σ or avoid it. The construction reveals thepossibility of further constraints as well as the dynamics being not completely determined .In the next lecture, we will consider a suitable classification of constraints and obtain acorresponding classification of constrained Hamiltonian systems.15ere are some elementary examples.1. H = p m , φ ( q ) = q − q : { φ, H } = pm ≈ p ≈
0, is a secondary constraint;2. H = p m , φ ( q ) = q : { φ, H } = qpm ≈ H = ap, a = 0 , φ ( q ) = q : { φ, H } = a ≈ hapter 4Dirac-Bergmann theory ofConstrained Hamiltonian Systems Our consistent constrained Hamiltonian system is specified by, H = H + k X a =1 λ a φ a , φ a ≈ , χ A ≈ , A = 1 , · · · l , k + l < n . (4.1)where ‘ ≈ ’ means evaluation on Σ defined by the primary constraints ( φ a ≈
0) and thesecondary constraints ( χ A ≈ { φ a , H } + { φ a , φ b } λ b ≈ , { χ A , H } + { χ A , φ b } λ b ≈ . (4.2)Thus we have k + l equations for k Lagrange multipliers and the system is naively, over-determined.Let rank of the ( k + l ) × k matrix of Poisson brackets of the constraints be K ≤ k . Thismeans that K is the maximum number of linearly independent rows and columns of thematrix. Thus there exist k − K independent relations among the k columns of the matrixi.e. ∃ ξ ( α ) b , α = 1 , · · · , k − K numbers such that { φ a , φ b } ξ ( α ) b ≈ , { χ A , φ b } ξ ( α ) b ≈ . (4.3)Now define two sets of linear combinations of the primary constraints namely,˜ φ α := ξ ( α ) a φ a , ˜ φ α ′ := η ( α ′ ) a φ a , α = 1 , · · · , k − K and α ′ = 1 , · · · , K . (4.4)The new set of K vectors η ( α ′ ) a are so chosen that the the set of constraints ˜ φ α , ˜ φ α ′ arefunctionally independent. Now it is clear that { φ a , ˜ φ α } ≈ , { χ A , ˜ φ α } ≈ . (4.5)Thus the k − K new combinations, ˜ φ α of primary constraints have a weakly vanishingPoisson bracket with all the constraints. Such constraints are termed first class constraints .Constraints which do not have this property are termed second class constraints . With thehelp of the ξ ’s we have regrouped the primary constraints into primary first class and primarysecond class constraints. 17e would like to do the same for secondary constraints. Observe that linear combination ofsecond class constraints will again satisfy the consistency conditions and to such combinationscould be added any combination of primary constraints without affecting these equations.To maintain the functional independence of the secondary constraints we need to ensure thatthe linear combinations are also functionally independent. Thus, consider the combinations,˜ χ A := S AB χ B + S Aα φ α + S Aα ′ φ α ′ (4.6)with S ·· so chosen that S AB is a non-singular matrix and ˜ χ A have weakly vanishing Poissonbracket with all constraints for a maximal number of values of A . Let this number be L ≤ l .Thus, we divide the combinations ˜ χ into the first class combinations, ˜ χ σ , σ = 1 , · · · , L andthe second class combinations ˜ χ σ ′ , σ ′ = 1 , · · · , l − L .The result of these manipulations is that (a) we can write λ a φ a = ˜ λ α ˜ φ α + ˜ λ α ′ ˜ φ α ′ and (b) theconsistency conditions can be simplified as, { ˜ φ α , H } ≈ , { ˜ χ σ , H } ≈ { ˜ φ α ′ , H } + ˜ λ β ′ { ˜ φ α ′ , ˜ φ β ′ } ≈ , { ˜ χ σ ′ , H } + ˜ λ β ′ { ˜ χ σ ′ , ˜ φ β ′ } ≈ . (4.8)The first set of equations involve only the k − K + L first class constraints and no Lagrangemultipliers while the second set of K + l − L equations involve only the K Lagrange mul-tipliers, ˜ λ α ′ . The k − K Lagrange multipliers, ˜ λ α have dropped out of the equations andwill remain undetermined . Once again we have more equations than unknown, but becauseof the separation into first and second class constraints, we are now guaranteed that the( K + l − L ) × K matrix of Poisson brackets of ˜ φ α ′ and ˜ χ σ ′ with ˜ φ β ′ has the maximal rank K .For, if it did not, there would exist further linear combinations which will weakly Poissoncommute with all constraints and by construction we have obtained the maximum numberof first class constraints. We will now solve for the Lagrange multipliers ˜ λ α ′ explicitly.Define the matrix ∆ of Poisson brackets of the second class constraints as,∆ := (cid:18) { ˜ φ α ′ , ˜ φ β ′ } { ˜ φ α ′ , ˜ χ σ ′ }{ ˜ χ ρ ′ , ˜ φ β ′ } { ˜ χ ρ ′ , ˜ χ σ ′ } (cid:19) := (cid:18) ∆ α ′ β ′ ∆ α ′ σ ′ ∆ ρ ′ β ′ ∆ ρ ′ σ ′ (cid:19) (4.9)This is an antisymmetric square matrix of order K + l − L . This must be non-singular .For, if it is singular, there will exist non-trivial linear combination of the second constraints˜ φ α ′ , ˜ χ σ ′ , which will weakly Poisson commute with all the second class constraints (and itautomatically commutes with the first class constraints), implying that we have additionalfirst class constraint. The non-singularity also requires that the total number of second classconstraints, K + l − L , must be an even integer . Let its (weak) inverse be the matrix C , C := (cid:18) C α ′ β ′ C α ′ σ ′ C ρ ′ β ′ C ρ ′ σ ′ (cid:19) (4.10)The equation C ∆ ≈ translates into, C α ′ β ′ ∆ β ′ γ ′ + C α ′ σ ′ ∆ σ ′ γ ′ = δ γ ′ α ′ (4.11) C ρ ′ β ′ ∆ β ′ τ ′ + C ρ ′ σ ′ ∆ σ ′ τ ′ = δ τ ′ ρ ′ (4.12) C α ′ β ′ ∆ β ′ τ ′ + C α ′ σ ′ ∆ σ ′ τ ′ = 0 (4.13) C ρ ′ β ′ ∆ β ′ γ ′ + C ρ ′ σ ′ ∆ σ ′ γ ′ = 0 (4.14)18he equations for the Lagrange multipliers become, { ˜ φ β ′ , H } + ∆ β ′ γ ′ ˜ λ γ ′ ≈ , { ˜ χ σ ′ , H } + ∆ σ ′ γ ′ ˜ λ γ ′ ≈ C α ′ β ′ , second one by C α ′ σ ′ , add the two and use (4.11) to solve for˜ λ α ′ . One gets, ˜ λ α ′ ≈ − C α ′ β ′ { ˜ φ β ′ , H } − C α ′ σ ′ { ˜ χ σ ′ , H } (4.16)Similar manipulation using (4.14) equation leads to, C ρ ′ β ′ { ˜ φ β ′ , H } + C ρ ′ σ ′ { ˜ χ σ ′ , H } ≈ H . We will write itin a more convenient form. ddt f ( ω ( t )) ≈ { f, H } + λ α { f, ˜ φ α }−{ f, ˜ φ α ′ } C α ′ β ′ { ˜ φ β ′ , H } − { f, ˜ φ α ′ } C α ′ σ ′ { ˜ χ σ ′ , H } (4.18) −{ f, ˜ χ ρ ′ } C ρ ′ β ′ { ˜ φ β ′ , H } − { f, ˜ χ ρ ′ } C ρ ′ σ ′ { ˜ χ σ ′ , H } The last line, which is weakly zero due to (4.17), has been added to get a more symmetricfinal expression.Now let us denote all the second class constraints, ( ˜ φ α ′ , ˜ χ ρ ′ ) by ξ m , m = 1 , · · · , K + l − L . Then,the nonsingular matrix ∆ is just the matrix ∆ mn = { ξ m , ξ n } , C mn is its weak inverse as beforeand the last group of four terms in (4.18) are conveniently expressed as −{ f, ξ m } C mn { ξ n , H } so that finally we obtain, ddt f ( ω ( t )) ≈ { f, H } + X α λ α { f, φ α } − { f, ξ m } (∆ − ) mn { ξ n , H } , (4.19): ≈ { f, H + X α λ α φ α } ∗ where,∆ mn := { ξ m , ξ n } and (4.20) { f, g } ∗ := { f, g } − { f, ξ m } (∆ − ) mn { ξ n , g } (Dirac Bracket) . (4.21)In this final expression, we have removed the ˜, the primary first class constraints are denotedby φ α while all second class constraints are denoted by ξ m .Several remarks are in order.1. The first step in the analysis of constrained systems was to obtain the full set ofconstraints starting with a given Hamiltonian H and a set of primary constraintsdefining the constraint surface Σ ⊂ Γ. In order to ensure that we get the final form ofevolution equations to be a Hamiltonian form, we used a modified Hamiltonian H andthought of the system as thought it were un-constrained in the sense the variations ofthe phase space coordinates were independent . To make contact with the constrainednature of the system, we required that the un-constrained dynamics be such that itstrajectories either lie in Σ or never intersect it. This lead us to discovering possible,additional constraints. Note that restrictions on the trajectories was with reference to19he sub-manifold Σ and hence only the primary constraints played a role in subsequentanalysis i.e. we did not add to the Hamiltonian, terms corresponding to the secondaryconstraints. The secondary constraints however do reveal that the required segregationof trajectories holds only with respect Σ ′ ⊂ Σ, defined by vanishing of all constraints.2. The next step was essentially an exercise in linear algebra. We did this to solve ex-plicitly for those Lagrange multipliers which could be solved for. This was facilitatedby regrouping the set of all constraints into first class and second class constraints.The final result reveals that evolution could be arbitrary if there are primary, firstclass constraints due to the undetermined λ α . The most compact expression for theHamiltonian evolution was obtained using the Dirac brackets .3. We now define first class variables as those functions on Γ whose Poisson brackets with all constraints are weakly zero. By the consistency condition (4.2), the Hamiltonian H is a first class variable and of course so are the first class constraints. The Hamiltonian H may or may not be a first class variable. It’s Poisson bracket with first classconstraints is of course weakly zero from (4.7). It is easy to check that sums andproducts of first class variables is again first class and so are the Poisson brackets offirst class variables .4. As noted already, in the presence of primary first class constraints, evolution equationfor a generic function f , contains the arbitrary Lagrange multipliers, λ α . From (4.19),it follows that evolution of first class variables is independent of λ α and is entirelygoverned by H . This justifies why first class variables are singled out.5. The Dirac brackets have been introduced as a convenient compact notation. Howeverit has many interesting properties, namely,(a) { f, g + h } ∗ ≈ { f, g } ∗ + { f, h } ∗ (addition);(b) { f, µg } ∗ ≈ µ { f, g } ∗ (scalar multiplication);(c) { f, gh } ∗ ≈ { f, g } ∗ h + g { f, h } ∗ (Leibniz);(d) { f, g } ∗ ≈ − { g, f } ∗ (antisymmetry);(e) { f, { g, h } ∗ } ∗ + cyclic ≈ { f, g } ∗ ≈ { f, g } for any first class f and ∀ g ;(b) { ξ i , g } ∗ ≈ , ∀ g and any second class constraint ξ i .Recall that weak equations involving Poisson brackets mean that the Poisson bracketsare first computed in a neighbourhood of Σ and then evaluated on
Σ. This rule isnecessary since a weakly vanishing function need not have a weakly vanishing Poissonbracket (since some of the partial derivatives ‘off’ Σ may not be zero). This applies tothe second class constraints as well. However, Dirac bracket of a second class constraintwith any function is weakly zero. Therefore, if we use Dirac brackets for writing theequations of motion (as shown in (4.19)), then we can set the second class constraints tobe zero before computing the Dirac brackets. This is equivalent to reducing the phase Functions on the phase space of an unconstrained system are generally called observables while in thecontext of a gauge system, the first class observables are all called as
Dirac observables . n to (2 n - the number of second class constraints). Thus, secondclass constraints correspond to redundant degrees of freedom which can be ignored bysetting them to zero .6. We could now focus on systems that do not have any second class constraints either apriori or after eliminating them via the Dirac bracket procedure. One is effectively leftsystems with only first class constraints. As noted earlier, the evolution of a dynamicalvariable in such systems is in general, arbitrary and only variables whose evolution is not arbitrary are the first class variables for whom the Dirac brackets are same as thePoisson brackets.Note that it could happen that there are no first class constraints left. In such a case,we have just the usual types of systems. Thus, the net conclusion of the analysis is:Generically, consistent Hamiltonian systems are systems with first class constraintswith the special case of no first class constraints. Hamiltonian systems with at least onefirst class constraint are termed gauge theories . We will now focus on these exclusively.Let Γ be a 2 n dimensional phase on which is given a first class Hamiltonian function H anda set of of k < n first class constraints, φ a whose vanishing defines the constraint surfaceΣ. The total Hamiltonian governing time evolutions is given by H := H + P a λ a φ a . Byvirtue of being first class, we have the following relations: { φ a , φ b } ≈ ↔ { φ a , φ b } = C abc ( ω ) φ c and { H , φ a } ≈ ↔ { H , φ a } = D ab ( ω ) φ b . (4.22)The evolution equation for any function f on Γ is given by, ddt f ( ω ( t )) ≈ { f ( ω ) , H + X a λ a φ a ( ω ) }| ω = ω ( t ) (4.23)Now observe that (a) if we make an arbitrary diffeomorphism i.e. a mapping of the manifoldΓ on to itself preserving differential structure, the manifold is unchanged (by definition).However, the symplectic form would change in general; (b) if we specialize the diffeomor-phisms to those which preserve the symplectic form, then the restricted diffeomorphisms arethe familiar canonical transformations . All such (continuously connected to identity) trans-formations can be generated by arbitrary functions on Γ, by the rule: δ ǫg ω µ := ǫ Ω µν ∂ ν g ( ω ).All of these however do not preserve the constraint surface; (c) To preserve the constraintsurface, the function must preserve the constraints defining the surface i.e. must be a firstclass function. Thus, all first class functions and in particular the first class constraints, dopreserve Σ. While constraints preserve the Hamiltonian, other first class functions need not.Those first class functions which do preserve the Hamiltonian as well are said to generate symmetry transformations . By contrast, the transformations generated by the first classconstraints are distinguished as gauge transformations .Thus, the diffeomorphisms generated by first class constraints, preserve the entire structure ofthe constrained Hamiltonian system (i.e. manifold, symplectic structure, constraint surfaceand the Hamiltonian) and are termed gauge transformations . First class functions (whichare not constraints) are automatically invariant under these transformations. Provided theyPoisson commute with the Hamiltonian, they generate symmetry transformations . Thisdistinction among the set of all first class function, comes about for the following reason.21ecall that λ a are undetermined, arbitrary functions that appear in the equations of motions.Therefore, beginning from any initial condition ˆ ω ∈ Σ, the actual trajectories will dependon the choice made for the λ a ’s. If we identified points on Σ as representing physical statesof the system, we would loose determinism – a given state does not uniquely determine thefuture state. We need to identify physical states of the system differently.Consider infinitesimal evolutions for two different choices of λ a ’s. We will have, ω ′ ( δt ) = ˆ ω + δt { ω, H ( λ ′ , ω ) }| ˆ ω , ω ( δt ) = ˆ ω + δt { ω, H ( λ, ω ) }| ˆ ω , (4.24)which implies that, δω ( δt ) = δt { ω, X a δλ a φ a }| ˆ ω = X a ( δtδλ a ) { ω, φ a }| ˆ ω , (4.25)which is nothing but the infinitesimal transformation generated by the first class constraints!Thus, if we arbitrarily choose the λ a and consider a trajectory evolved from some ˆ ω thenanother trajectory evolved by a different choice of λ a from the same initial point, would beobtained by making a gauge transformation. Clearly, if we identified points in Σ which arerelated by gauge transformations as being ‘physically the same’, then we regain determin-ism in the sense that physical states evolve into unique physical states. Thus the apparentdynamical indeterminism implied by the first class constraints appearing in the Hamiltonian,can be resolved by defining equivalence classes of points of Σ under transformations generatedby the constraints (i.e. gauge transformations) to represent the physical states of the system. Notice that this identification of physical states with equivalence classes under gauge trans-formations involves only those first class constraints which appear in the Hamiltonian sinceonly these have a bearing on the dynamical evolution.Since there are k first class constraints, the set of points of Σ which are related by gaugetransformations is parameterized by k parameters and hence the set of gauge equivalenceclasses is parameterized by 2 n − k − k parameters. This space is called the Reduced PhaseSpace . This space be made explicit by introducing additional k ‘constraints’ (now calledas gauge fixing conditions – χ a ( ω ) ≈ φ a , χ b constraints is non-singular. Demanding their preservation intime fixes the Lagrange multipliers and hence the name.Having clarified the identification of physical states, the definition of physical observablesand notions of symmetry obviously must be formulated for physical states. The observablesmust have unique evolutions since by definition these are supposed to be functions of physicalstates. Only first class functions satisfy this property and only these qualify to be termed asphysical observables. Likewise, notion of symmetry should refer to invariance with respectto transformations of physical states , the generators of infinitesimal symmetries must mapthe entire gauge equivalence classes among themselves and of course preserve the evolution.Clearly these again have to be first class functions and must additionally Poisson commutewith the Hamiltonian.To summarize:1. Hamiltonian systems with at least one first class constraint, require identification ofphysical states not with individual points of the constraint surface but with gauge equiv-alence classes of points of the constraint surface.22. In view of the above, it is common to refer to the original phase space as the kinematicalphase space , Γ kin . The constraint sub-manifold of Γ kin is Σ. The physical state space(or reduced phase space) is denoted as Γ phys := Σ / ∼ where ∼ refers to the gaugetransformations. Note that the physical state space is not a sub-manifold of Σ.3. Although both the constraints and ‘conserved quantities’ serve to confine the tra-jectories the two are distinguished by the fact that constraint impose limitation onpossible initial conditions (restriction to Σ) as well as force a non-trivial identificationof physical states to ensure determinism of dynamics. Notions of conserved quantities,symmetries become meaningful only after this identification. Also conserved quantitiesdo not impose any ab initio limitation on the possible initial conditions but only on asubsequent trajectory.Observe that in the light of the discussion of symmetry and gauge transformations, a firstclass Hamiltonian H generates a symmetry transformation , namely time translations. How-ever there are theories in which H = 0 and H is made up entirely of first class constraints.Now the ‘time evolution’ itself becomes a gauge transformation and hence ‘no evolution’ ofphysical states. Next chapter discusses this case.23 hapter 5Systems with the Hamiltonian as aconstraint Consider special types of gauge theories in which H = 0 i.e. the Hamiltonian is entirely madeup of first class constraints. The prime physical example of such a system is the dynamics ofinhomogeneous cosmological space-times within the context of Einstein’s theory of GeneralRelativity .As is well known, Einstein’s general relativistic theory of gravity (GR) has a four dimensionalmanifold on which is defined some metric tensor (of Minkowskian signature) field whichmakes it in to a space-time. The metric tensor is not a fixed entity, as in the case ofspecial relativity (the Minkowski space-time), but is a dynamical entity i.e. is determinedin conjunction with the (interacting) matter distribution on the manifold. The equationdetermining the metric is the Einstein equation, which is a set of 10, local, partial differentialequations of order 2, for the 10 components of the metric tensor, g µν . It is non-trivial factthat these equations admit a well-posed initial value formulation i.e. (i) one can take the 4dimensional manifold as R × Σ , (ii) specify two symmetric tensor fields, g ij , and its timederivative, ˙ g ij on Σ , then the space-time can be determined for other times provided the‘initial data’ satisfies certain conditions. Furthermore, the system of equations can be cast inthe form of constrained Hamiltonian system, with Hamiltonian given entirely by first classconstraints. There are 4 sets of constraints (per point, since one is dealing with a field theory),three of which, called vector or diffeomorphism constraints and the remaining one called the scalar or Hamiltonian constraint . The vector constraints generate usual diffeomorphismsof Σ while the scalar constraint generates evolution of the ‘spatial manifold’, Σ in the4-manifold constructing a solution (space-time) of the Einstein equation. The upshot isthat solution space-times of Einstein equation can be viewed as phase space Hamiltoniantrajectories in the gravitational phase space with initial data satisfying a set of first classconstraints and with the Hamiltonian given as linear combination of these constraints. Theconstraints in the Hamiltonian formulation of general relativity, reflect the 4-diffeomorphisminvariance of GR [4].This is pretty complicated to deal with in general, however a simplification is possible. Ifwe restrict ourselves do the dynamics of only a special class of 3-geometries, namely, thosemetric tensors whose dependence on spatial coordinates (coordinates on Σ ) is completely For space-times corresponding to compact objects, typically asymptotically flat space times, there is a‘true Hamiltonian’, the analogue of H , generating asymptotic time translations. single Hamiltonianconstraint remaining. We will not need to take any specific model to illustrate the issues.We can also construct systems in which Hamiltonian is the single (and hence first class)constraint. To see this, let us begin with a usual un-constrained
Hamiltonian system with aphase space ˆΓ and a Hamiltonian H ( ω ). Let us extend this phase space to Γ by adding twomore conjugate variables, τ, π . Let φ ( τ, π, ω ) := H ( ω ) − π be chosen as the Hamiltonian onΓ and take it as a constraint as well, i.e. H := λφ ≈
0. Clearly, any evolution (with respectto T ) generated by the Hamiltonian is a gauge transformation and therefore unphysical.Functions which are insensitive to the evolution are the first class ones (Dirac observables), f . Let us look for Dirac observables. { f, H } ≈ ⇒ { f ( τ, π, ω ) , H ( ω ) } − ∂f∂τ ≈ . (5.1)where we have used the definition of Poisson bracket in the second equation.It is clear that functions which are independent of τ, π and satisfying the usual un-constrainedevolution equation in ˆΓ, are Dirac observables of the extended constrained dynamics. Func-tions depending only on τ are not Dirac observables while those depending only on π areDirac observables. Constants with respect to the τ -dynamics, also are Dirac observables.With this construction, we see that usual unconstrained dynamics can be viewed (albeittrivially) as a constrained dynamics in a bigger phase space. Furthermore, all functions on theunconstrained phase space, evolving by the un-constrained dynamics are Dirac observablesof the constrained dynamics. The Dirac observables however are constants with respect tothe T -evolution.Consider now the trajectories of the constrained dynamics. One finds, ddT τ = − λ , ddT π = 0 , ddT ω = λ { ω, H ( ω ) } = λ ∂∂τ ω . (5.2)The last equality is deduced from (5.1) with f = ω . We can define a new ‘time’, T ′ by theequation dT ′ = λdT , so that ddT ′ τ = − , ddT ′ π = 0 , ddT ′ ω = ∂∂τ ω . (5.3)Note that T evolution is generated by λφ while T ′ evolution is generated by φ . Startingwith some initial values at T ′ = 0, one generates the T ′ -trajectories. All the points alongthese are related by gauge transformations generated by φ . Thus the equivalence classesare in one-to-one and on-to correspondence with the points τ = 0 , π = ˆ π, ω = ˆ ω . Thegauge orbits which lie on the constraint surface satisfy ˆ π = H (ˆ ω ) and hence these orbits arecompletely determined by points in ˆΓ. Thus the reduced phase space in this case is just theun-constrained phase space ˆΓ.This simple construction brings out a few points. There are two notions of ‘evolution’ (i)the T or ( T ′ ) evolution, which is some times called an external time evolution and (ii) the τ evolution which is correspondingly called an internal time evolution . From the point of viewof the constrained system, τ is just one of the degrees of freedom which is singled out because25he constraint had a particularly simple additive form. The τ evolution can thus be thoughtof as evolution of a set of degrees of freedom with respect to a singled out degree of freedom.The internal time evolution is thus also called a relational evolution while the singled outdegree of freedom is called a clock degree of freedom . The arbitrary function, λ is also calleda lapse function and its arbitrariness corresponds to the freedom of re-parameterizing theexternal time. The Hamiltonian systems resulting from homogeneous cosmologies of GR,typically get presented in the form of the constrained system (the constraint however has adifferent form in general). The terminology used above is inherited from the GR context. Onecan in fact do a more general and systematic analysis of the notions of external and internaldynamics which my student Golam Hossain and I have carried out for finite dimensionalsystems.While classically, constrained Hamiltonian systems are interesting in their own right, theybecome more challenging at the quantum level. As all the fundamental interactions of stan-dard model and its extensions are gauge (field) theories, one has to face these challenges. Inthe perturbative analysis, one needs to ‘fix a gauge’, in order that propagators for gauge fieldscan be defined and then has to show that the final observable scattering cross-sections areindeed gauge invariant. When a quantum theory of gravity is attempted, the understandingof semiclassical approximation becomes quite complicated especially in a non-perturbativeapproach. 26 hapter 6Exercises
1. Check that the coordinate transformations generated by Hamiltonian vector fields leavethe symplectic form invariant .2. Check that (1.3) property is needed to prove the Jacobi identity for Poisson brackets.3. For the Maxwell theory, check that the secondary constraint holds for all times without having to require any further constraint. Furthermore, the Poisson bracket of the twoconstraints is also zero.4. Show that the Hamilton’s equations of motion can be identified with the Maxwellequations.5. For the Maxwell theory, we have already shown that the canonical transformations gen-erated by the first class constraints are indeed the usual gauge transformations (hencein fact the name). Show, by direct computation, that F µν are first class functions.6. Consider a massive, relativistic particle with action S = m R dτ p η µν ˙ x µ ˙ x ν . Carry outthe constraint analysis and give examples of first class functions.7. Consider the four dimensional phase space with coordinates ( q , q , p , p ). Considertwo constraints φ ( q, p ) := p + p + ( q ) + ( q ) − R and χ ( q, p ) := p . Let H ( q, p ) besome suitable Hamiltonian (not given explicitly) such that these two constraints arepreserved [5].(a) Identify the constraint surface.(b) Compute the expression for the Dirac bracket.27 ibliography [1] J. G. Papastavridis, Analytical Mechanics: A Comprehensive Treatise on Dynamics ofConstrained Systems , Oxford, 2002[2] P. A. M. Dirac,
Lectures on Quantum Mechanics , Yeshiva University Press, New York,1964;[3] G. Sudarshan and N. Mukunda,
Classical Dynamics: A Modern Perspective , chap. 8, 9,[4] R. M. Wald,
General Relativity , The University of Chicago Press, 1984, especially,chapter 10 and appendix E.[5] Radhika Vathsan,
J. Math. Phys.37