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Chapter 8 Field Theory In section 5.4 we considered the continuum limit of a chain of point masses on stretched string. We had a situation in which the poten- tial energy had interaction terms for particle A which depended only on the relative displacements of particles in the neighborhood of A. If we take our coordinates to be displacements from equilibrium, and consider only motions for which the displacement η = η(x, y, z, t)be- comes differentiable in the continuum limit, then the leading term in the potential energy is proportional to second derivatives in the spacial coordinates. For our points on a string at tension τ,withmassdensity ρ, we found T = 1 2 ρ L 0 ˙y 2 (x)dx, U = τ 2 L 0 ∂y ∂x 2 dx, as we can write the Lagrangian as an integral of a Lagrangian density L(y, ˙y, y ,x,t). Actually for our string we had no y or x or t dependence, because we ignored gravity U g = ρgy(x, t)dx, and had a homogeneous string whose properties were time independent. In general, however, such dependence is quite possible. For a three dimensional object, such as the equations for the displacement of the atoms in a crystal, we might have fields η, the three components of the displacement of a particle, as a function of the three coordinates (x, y, z) determining the particle, 219 220 CHAPTER 8. FIELD THEORY as well as time. Thus the generalized coordinates are the functions η i (x, y, z, t), and the Lagrangian density will depend on these, their gradients, their time derivatives, as well as possibly on x, y, z, t.Thus L = L(η i , ∂η i ∂x , ∂η i ∂y , ∂η i ∂z , ∂η i ∂t ,x,y,z,t) and L = dx dy dz L, I = dx dy dz dt L. The actual motion of the system will be given by a particular set of functions η i (x, y, z, t), which are functions over the volume in ques- tion and of t ∈ [t I ,t f ]. The function will be determined by the laws of dynamics of the system, together with boundary conditions which depend on the initial configuration η I (x, y, z, t I ) and perhaps a final configuration. Generally there are some boundary conditions on the spacial boundaries as well. For example, our stretched string required y =0atx =0andx = L. Before taking the continuum limit we say that the configuration of the system at a given t was a point in a large N dimensional configura- tion space, and the motion of the system is a path Γ(t) in this space. In the continuum limit N →∞, so we might think of the path as a path in an infinite dimensional space. But we can also think of this path as a mapping t → η(·, ·, ·,t) of time into the (infinite dimensional) space of functions on ordinary space. Hamilton’s principal says that the actual path is an extremum of the action. If we consider small variations δη i (x, y, z, t) which vanish on the boundaries, then δI = dx dy dz dt δL =0. Note that what is varied here are the functions η i , not the coordinates (x, y, z, t). x, y, z do not represent the position of some atom — they represent a label which tells us which atom it is that we are talking about. They may well be the equilibrium position of that atom, but 221 they are independent of the motion. It is the η i which are the dynamical degrees of freedom, specifying the configuration of the system. The variation δL(η i , ∂η i ∂x , ∂η i ∂y , ∂η i ∂z , ∂η i ∂t ,x,y,z,t) = ∂L ∂η δη + ∂L ∂(∂η/∂x) δ ∂η ∂x + ∂L ∂(∂η/∂y) δ ∂η ∂y + ∂L ∂(∂η/∂z) δ ∂η ∂z + ∂L ∂(∂η/∂t) δ ∂η ∂t . Notice there is no variation of x, y, z,andt, as we discussed. The notation is getting awkward, so we need to reintroduce the notation A ,i = ∂A/∂r i .Infact,weseethat∂/∂t enters in the same way as ∂/∂x, so we will set x 0 = t and write ∂ µ := ∂ ∂x µ = ∂ ∂t , ∂ ∂x , ∂ ∂y , ∂ ∂z , for µ =0, 1, 2, 3, and write η ,µ := ∂ µ η. If there are several fields η i ,then ∂ µ η i = η i,µ . The comma represents the beginning of differentiation, so we must not use one to separate different ordinary indices. In this notation, we have δL = i ∂L ∂η i δη i + i 3 µ=0 ∂L ∂η i,µ δη i,µ , and δI = i ∂L ∂η i δη i + i 3 µ=0 ∂L ∂η i,µ δη i,µ d 4 x, where d 4 x = dx dy dz dt. Except for the first term, we integrate by parts, δI = i ∂L ∂η i − i 3 µ=0 ∂ µ ∂L ∂η i,µ δη i d 4 x, 222 CHAPTER 8. FIELD THEORY where we have thrown away the boundary terms which involve δη i eval- uated on the boundary, which we assumed to be zero. Inside the region of integration, the δη i are independent, so requiring δI =0forall functions δη i (x µ ) implies ∂ µ ∂L ∂η i,µ − ∂L ∂η i =0. (8.1) We have written the equations of motion (which is now a partial dif- ferential equation rather than coupled ordinary differential equations), in a form which looks like we are dealing with a relativistic problem, because t and spatial coordinates are entering in the same way. We have not made any assumption of relativity, however, and our problem will not be relativistically invariant unless the Lagrangian density is invariant under Lorentz transformations (as well as translations). Now consider how the Lagrangian changes from one point in space- time to another, including the variation of the fields, assuming the fields obey the equations of motion. Then the total derivative for a variation of x µ is dL dx µ = ∂L ∂x µ η + ∂L ∂η i η i,µ + ∂L ∂η i,ν η i,ν,µ . Plugging the equations of motion into the second term, dL dx µ = ∂L ∂x µ + ∂ ν ∂L ∂η i,ν η i,µ + ∂L ∂η i,ν η i,µ,ν = ∂L ∂x µ + ∂ ν ∂L ∂η i,ν η i,µ . Thus ∂ ν T µν = − ∂L ∂x µ , (8.2) where the stress-energy tensor T µν is defined by T µν (x)= ∂L ∂η i,ν η i,µ −Lδ µν . (8.3) 223 Note that if the Lagrangian density has no explicit dependence on the coordinates x µ , the stress-energy tensor satisfies an equation ∂ ν T µν which is a continuity equation. In dynamics of discrete systems we defined the Hamiltonian as H = i p i ˙q i − L(q, p, t). Considering the continuum as a limit, L = d 3 xL is the limit of ijk ∆x∆y∆zL ijk ,whereL ijk depends on q ijk and a few of its neighbors, and also on ˙q ijk . The conjugate momentum p ijk = ∂L/∂ ˙q ijk =∆x∆y∆z∂L ijk /∂ ˙q ijk which would vanish in the continuum limit, so instead we define π(x, y, z)=p ijk /∆x∆y∆z = ∂L ijk /∂ ˙q ijk = δL/δ ˙q(x, y, z). The Hamiltonian H = p ijk ˙q ijk − L = ∆x∆y∆zπ(x, y, z)˙q(xyz) − L = d 3 x (π(r)˙q(r) −L)= d 3 xH, where the Hamiltonian density is defined by H(r)=π(r)˙q(r) −L(r). Of course if there are several fields q i at each point, H(r)= i π i (r)˙q i (r) −L(r). where π i (r)= δL δ ˙q i (r) . Notice that the Hamiltonian density is exactly T 00 , one component of the stress-energy tensor. Consider the case where L does not depend explicitly on (x, t), so 3 ν=0 ∂ ν T µν =0, or ∂ ∂t T µ0 = 3 i=1 ∂ i T µi =0. 224 CHAPTER 8. FIELD THEORY This is a continuity equation, similar to the equation from fluid me- chanics, ∂ρ/∂t + ∇·(ρv) = 0, which expresses the conservation of mass. That equation has the interpretation that the change in the mass contained in some volume is equal to the flux into the volume, because ρv is the flow of mass past a unit surface area. In the current case, we have four conservation equations, indexed by µ.Eachofthese can be integrated over space to tell us about the rate of change of the “charge” Q µ (t)= d 3 VT µ0 (x, t), d dt Q µ (t)= d 3 V ∂ ∂x i T µi (x, t). We see that his is the integral of the divergence of a vector current ( J µ ) i = T µi , which by Gauss’ law becomes a surface integral of the flux of J µ out of the volume of our system. We have been sloppy about our boundary conditions, but in many cases it is reasonable to assume there is no flux out of the volume. In this case the right hand side vanishes, and we find four conserved quantities Q µ (t)=constant. For µ =0wesawthatT 00 is the energy density, so Q 0 is the total energy. Cyclic coordinates In discrete mechanics, when L was independent of a coordinate q i ,even though it depended on ˙q i , we called the coordinate cyclic or ignorable, and found a conserved momentum conjugate to it. For fields in general, L(η, ˙η, ∇η) depends on spatial derivates of η as well, and we may ask whether we need to require absense of dependence on ∇η for a coordi- nate to be cyclic. Independence of both η and ∇η implies independence on an infinite number of discrete coordinates, the values of η(r)atev- ery point r, which is too restrictive a condition for our discussion. We will call a coordinate field η i cyclic if L does not depend directly on η i , although it may depend on its derivatives ˙η i and ∇η i . The Lagrange equation then states µ ∂ µ δL δη i,µ =0, or d dt π i + j ∂ j δL δη i,j =0. 8.1. NOETHER’S THEOREM 225 If we integrate this equation over all space, and define Π i (t)= π i (r)d 3 r, then the derivative dΠ/dt involves the integral of a divergence, which by Gauss’ law is a surface term dΠ(t) dt = − δL δη i,j (dS) j . Assuming the spatial boundary conditions are such that we may ignore this boundary term, we see that Π i (t) is a constant of the motion. 8.1 Noether’s Theorem We want to discuss the relationship between symmetries and conserved quantities which is known as Noether’s theorem. It concerns in- finitesimal tranformations of the degrees of freedom η i (x µ )whichmay relate these to degrees of freedom at a changed point. That is, the new fields η (x ) is related to η(x) rather than η(x ), where x µ → x µ = x µ + δx µ is some infinitesimal transformation of the coordinates rather than of the degrees of freedom. For a scalar field, like temperature, under a rotation, we would define the new field η (x )=η(x), but more generally the field may also change, in a way that may depend on other fields, η i (x )=η i (x)+δη i (x; η k (x)). This is what you would expect for a vector field E under rotations, because the new E x gets a component from the old E y . The Lagrangian is a given function of the old fields L(η i ,η i,µ ,x µ ). If we substitute in the values of η(x)intermsofη (x ) we get a new function L , defined by L (η i ,η i,µ ,x µ )=L(η i ,η i,µ ,x µ ). 226 CHAPTER 8. FIELD THEORY The symmetries we wish to discuss are transformations of this type under which the form of the Lagrangian density does not change, so that L is the same functional form as L,or L (η i ,η i,µ ,x µ )=L(η i ,η i,µ ,x µ ). In considering the action, we integrate the Lagrangian density over a region of space-time between two spacial slices corresponding to an initial time and a final time. We may, however, consider an arbitrary region of spacetime Ω ⊂ R 4 . The corresponding four dimensional vol- ume in the transformed coordinates is the region x ∈ Ω . The action for a given field configuration η S(η)= Ω L(η, η ,µ ,x)d 4 x differs from S (η )= Ω L (η ,η ,µ ,x )d 4 x ) only by the Jacobian, as a change of variables gives S (η )= Ω ∂x ∂x L(η, η ,µ ,x)d 4 x. The Jacobian is det (δ µν + ∂ ν δx µ )=1+Tr ∂δx µ ∂x ν =1+∂ µ δx µ . It makes little sense to assume the Lagrangian density is invariant unless the volume element is as well, so we will require the Jacobian to be identically 1, or ∂ µ δx µ =0. SothenδS = 0 for the symmetries we wish to discuss. We can also consider S (x )asanintegraloverx,asthisisjusta dummy variable, S (η )= Ω L η (x),η ,µ (x),x d 4 x. This differs from S(η)byS (η ) −S(η)=δ 1 S + δ 2 S, because 1. the Lagrangian is evaluated with the field η rather than η,pro- ducing a change δ 1 S = δL δη i ¯ δη i + δL δη i,µ ¯ δη i,µ d 4 x, 8.1. NOETHER’S THEOREM 227 where ¯ δη i (x):=η i (x)−η i (x)=η i (x)−η i (x )+δη i (x)=δη i (x)−η i,µ δx µ . 2. Change in the region of integration, Ω rather than Ω, δ 2 S = Ω − Ω L(η, η ,µ ,x)d 4 x. If we define dΣ µ to be an element of the three dimensional surface Σ=∂Ω of Ω, with outward-pointing normal in the direction of dΣ µ , the difference in the regions of integration may be written as an integral over the surface, Ω − Ω d 4 x = Σ δx µ · dΣ µ . Thus δ 2 S = ∂Ω Lδx µ · dS µ = Ω ∂ µ (Lδx µ ) (8.4) by Gauss’ Law (in four dimensions). As ¯ δ is a difference of two functions at the same values of x,this operator commutes with partial differentiation, so ¯ δη i,µ = ∂ µ ¯ δη i .Using this in the second term of δ 1 S and the equations of motion in the first, we have δ 1 S = d 4 x ∂ µ ∂L ∂η i,µ ¯ δη i + ∂L ∂η i,µ ∂ µ ¯ δη i = Ω d 4 x∂ µ ∂L ∂η i,µ ¯ δη i = Ω d 4 x∂ µ ∂L ∂η i,µ δη i − ∂L ∂η i,µ η i,ν δx ν . Then δ 1 S + δ 2 S = 0 is a condition in the form Ω d 4 x∂ µ J µ =0, (8.5) 228 CHAPTER 8. FIELD THEORY which holds for arbitrary volumes Ω. Thus we have a conservation equation ∂ µ J µ =0. The infinitesimal variations may be thought of as proportional to an infinitesimal parameter , which is often in fact a component of a four- vector. The variations in x µ and η i are then δx µ = dx µ d ,δη i = dη i d , so if δ 1 S + δ 2 S =0is− times (8.5), J µ = − ∂L ∂η i,µ dη i d + ∂L ∂η i,µ η i,ν dx ν d −L dx µ d . = − ∂L ∂η i,µ dη i d + T νµ dx ν d . (8.6) Exercises 8.1 The Lagrangian density for the electromagnetic field in vacuum may be written L = 1 2 E 2 − B 2 , where the dynamical degrees of freedom are not E and B, but rather A and φ,where B = ∇×A E = − ∇φ − 1 c ˙ A a) Find the canonical momenta, and comment on what seems unusual about one of the answers. b) Find the Lagrange Equations for the system. Relate to known equations for the electromagnetic field. [...]... Math Methods of Classical Mechanics SpringerVerlag, New York, 1984 QA805.A6813 [3] R Creighton Buck Advanced Calculus McGraw-Hill, 1956 [4] Herbert Goldstein Classical Mechanics Addison-Wesley, Reading, Massachusetts, second edition, 1980 QA805.G6 [5] I S Gradshtein and I M Ryzhik Table of integrals, series, and products Academic Press, New York, 1965 QA55.R943 [6] L Landau and Lifschitz Mechanics Pergamon... QA805.L283/1976 [7] Jerry B Marion and Stephen T Thornton Classical Dynamics Harcourt Brace Jovanovich, San Diego, 3rd ed edition, 1988 QA845.M38/1988 [8] R A Matzner and L C Shepley Classical Mechanics Prentice Hall, Englewood Cliffs, NJ, 91 QC125.2.M37 ISBN 0-13-137076-6 [9] Morris Edgar Rose Elementary Theory of Angular Momentum Wiley, New York, 1957 QC174.1.R7 [10] Walter Rudin Principles of Mathematical Analysis... action-angle, 186 active, 88 adiabatic invariant, 210 angular momentum, 9 antisymmetric, 95 apogee, 72 apsidal angle, 76 associative, 91 attractor, 29 autonomous, 24 D’Alembert’s Principle, 42 diffeomorphism, 176 differential cross section, 82 differential k-form, 160 Dirac delta function, 144 dynamical balancing, 106 dynamical systems, 23 bac-cab, 77, 98, 232 body cone, 109 body coordinates, 86 Born-Oppenheimer,... 1953 241 242 BIBLIOGRAPHY [11] M Spivak Differential Geometry, volume 1 Publish or Perish, Inc., 1970 [12] Keith R Symon Mechanics Addsion-Wesley, Reading, Massachusetts, 3rd edition, 1971 QC125.S98/1971 ISBN 0-201-073927 [13] Eugene Wigner Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra Academic Press, New York, 1959 Index composition, 89 conditionally periodic motion, 192 configuration... Born-Oppenheimer, 131 eccentricity, 72 electrostatic potential, 57 elliptic fixed point, 32 enthalpy, 150 Euler’s equations, 108 Euler’s Theorem, 92 exact, 149, 164 extended phase space, 7, 171 exterior derivative, 163 exterior product, 161 canonical transformation, 153 canonical variables, 153 center of mass, 10 centrifugal barrier, 68 Chandler wobble, 113 closed, 165 closed under, 91 complex structure on phase space,... to the curl as to any ordinary vector, except that one must be careful not to change, by reordering, what the differential operators act on In particular, Eq A.7 is A×( × B) = Ai Bi − i Ai i ∂B ∂xi (B .10) 236 APPENDIX B THE GRADIENT OPERATOR Appendix C Gradient in Spherical Coordinates The transformation between Cartesian and spherical coordinates is given by 1 r= (x2 + y 2 + z 2 ) 2 x= r sin θ cos . =0wesawthatT 00 is the energy density, so Q 0 is the total energy. Cyclic coordinates In discrete mechanics, when L was independent of a coordinate q i ,even though it depended on ˙q i , we called. operators act on. In particular, Eq. A.7 is A ×( ∇× B)= i A i ∇B i − i A i ∂ B ∂x i . (B .10) 236 APPENDIX B. THE GRADIENT OPERATOR Appendix C Gradient in Spherical Coordinates The transformation
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