Vector Field Theory (E&M)

Transcription

1 Physis 4 Leture 2 Vetor Field Theory (E&M) Leture 2 Physis 4 Classial Mehanis II Otober 22nd, 2007 We now move from first-order salar field Lagrange densities to the equivalent form for a vetor field. Our model (and ultimate goal) is a desription of vauum eletrodynamis. From the natural ation, we derive the field equations Maxwell s equations for potentials in the absene of soures. Soureless field theories tell us nothing about the various onstants whih must appear in a general ation, oupling soures to the field theory density, but by appealing to our E&M bakground, we an set the various onstants appropriate to soure oupling. So we begin with the ation and field equations, then form the energymomentum tensor for the fields alone, and by setting our one overall onstant so that the T 00 omponent orresponds to energy density, we reover the full (familiar) energy-momentum tensor for E&M. After all of this, we are in good position to introdue harge and urrent soures. 2. Vetor Ation Our objetive is E&M, and our starting point will be an ation. The big hange omes in our Lagrange density rather than some simple L(φ, φ,µ, g µν ), we have a set of fields, all them A µ, so the ation an depend on salars made out of the fields A µ and their derivatives A µ,ν. Think of Minkowski spae, and our usual treatment of eletriity and magnetism. We know that the Field strength tensor F µν is a Lorentz-tensor, but why stop there? Why not make a full salar ation out of the field-strength tensor, and ignore its usual definition (we should reover this anyway, if our theory is properly defined)? To that end, the only important feature of F µν is its antisymmetry: F µν = F νµ. of 8

2 2.. VECTOR ACTION Leture 2 Sine we are shooting for a vetor field, a singly-indexed set of quantities, the seond rank tensor F µν must be assoiated, in first order form, with the momenta. The only salar we an make omes from double-ontration F µν F µν. So taking this as the Hamiltonian, our first-order ation has to look like S V = dτ g (F µν (derivatives of A µ ) 2 ) F µν g αµ g βν F αβ. (2.) Beause F µν is antisymmetri, we an form an antisymmetrization of A ν,µ, i.e. (A ν,µ A µ,ν ). As an interesting aside, we are impliitly assuming that the metri here is just the usual Minkowski, g µν = η µν, so that one might objet to the normal partial derivatives on the A µ,ν. This is atually not a problem if we use the antisymmetrized form, sine A ν;µ A µ;ν = ( A ν,µ Γ σ νµ A σ ) ( Aµ,ν Γ σ µν A σ ) = A ν,µ A µ,ν, using the symmetry (torsion-free) of the onnetion: Γ σ µν = Γ σ νµ. (2.2) So our proposed ation is S V = dτ g (F µν (A ν,µ A µ,ν ) 2 ) F µν g αµ g βν F αβ. (2.3) Again, sine this is first-order form, we vary w.r.t. F µν and A µ separately. Inidentally, the antisymmetry of F µν is now enfored beause it is ontrated with the antisymmetrization of A µ,ν, only the antisymmetri portion will ontribute, so we an vary w.r.t. all sixteen omponents independently. δs V δf µν =(A ν,µ A µ,ν ) g αµ g βν F αβ = 0 (2.4) and this gives a relation between F and A: F µν = A ν,µ A µ,ν. (2.5) The variation of the A µ is a little more diffiult with an obvious abuse of notation: δs V = dτ g (F µν (A ν,µ A µ,ν )) δa µ,ν A µ,ν = dτ g (A µ,ν (F νµ F µν )) δa µ,ν (2.6) A µ = dτ g (F νµ F µν ) δa µ,ν, 2 of 8

3 2.. VECTOR ACTION Leture 2 and we really have in mind the ovariant derivative (the g ensures that we mean what we say). We an use integration by parts to get a total divergene term, whih as usual an be onverted to a boundary integral where it must vanish, δs V = dτ [ g (F νµ F µν ) δa µ dτ ];ν [ g (F νµ F µν ) ] ;ν δa µ. }{{} =0 (2.7) The seond term s integrand must then vanish for arbitrary δa µ. Using ( g) ;ν = 0, the above gives the seond set of field equations: (F νµ F µν ) ;ν = 0 (2.8) and beause we know that F µν is antisymmetri (either a priori or from (2.5)), this redues to F µν ;ν = 0. (2.9) Combining these two, and writing in terms of A µ, we find that the fields A µ satisfy µ ν A ν ν ν A µ = 0. (2.0) If we are really in Minkowski spae with Cartesian spatial oordinates, this is a set of four independent equations whih we may write as 0 = [ A 0 ] + A [ 2 A 0 ] t t 2 t A 0 [ A 0 ] [ 0 = + A 2 ] (2.) A t 2 t A. Note the similarity with equations (0.4) and (0.5) from Griffiths : 2 V + t ( A) = ρ ɛ ( ) ( ) A V A µ 0 ɛ 0 t 2 A + µ 0 ɛ 0 = µ 0 J. t (2.2) It appears reasonable to interpret A 0 = V and A j = A, the eletri and magneti vetor potentials respetively. To proeed, we know that there Griffiths, David J. Introdution to Eletrodynamis. Prentie Hall, 999. See page 47 of the third edition. 3 of 8

4 2.. VECTOR ACTION Leture 2 must be gauge freedom the physial effets of the fields are transmitted through F µν. We know this to be true, although we have not yet done anything to establish it here, there being no soures in our theory. The gauge freedom is expressed in the onnetion between the field strength tensor and the potential If we take A µ A µ = A µ + ψ,µ, then F µν = A ν;µ A µ;ν = A ν,µ A µ,ν. (2.3) F µν = A ν,µ A µ,ν =(A ν,µ + ψ,νµ ) (A µ,ν + ψ,µν ) = A ν,µ A µ,ν, (2.4) and nothing hanges. We an exploit this to set the total divergene of A µ. Suppose we start with a set A µ suh that µ A µ = F (x), then we introdue a four-gradient: A µ + µ ψ, the new divergene is µ (A µ + µ ψ) = F + µ µ ψ (2.5) and if we like, this new divergene an be set to zero by appropriate hoie of ψ (i.e. µ µ ψ = F, Poisson s equation for the D Alembertian operator). Then we an a priori set µ A µ = 0 and the field equations an be simplified (from (2.0)) µ ν A ν ν ν A µ = ν ν A µ (2.6) whih beomes, under our identifiation A 0 = V : 0 = 2 2 V t V 0 = 2 2 A t A, (2.7) appropriate to soure-free potential formulation of E&M in Lorentz gauge. 2.. The Field Strength Tensor Now we an go on to define the omponents of the field strength tensor, but we are still laking an interpretation in terms of E and B. That s fine, for now, we an use our previous experiene as a ruth, but to reiterate we annot say exatly what the fields are (the independent elements of F µν ) until we have a notion of fore. 4 of 8

5 2.2. ENERGY-MOMENTUM TENSOR FOR E&M Leture 2 From F µν = ν A µ µ A ν, we have 2 0 F µν = ( A x t + V ) ( ) x A y t + V y 0 A y x Ax y ( A z t A z x A 0 z y 0 with representing the obvious antisymmetri entry. + V ) z Ax z Ay z, (2.8) If we take the typial definition E = V A t and B = A, then the field strength tensor takes its usual form F µν = E 0 x E y E z Ex 0 B z B y Ey B z 0 B x Ez B y B x 0. (2.9) 2.2 Energy-Momentum Tensor for E&M We now understand the Lagrangian density (2.3) as the appropriate integrand of the ation for the eletromagneti potential, or at least, the soure-free E&M form. Our next immediate task is to identify the energymomentum tensor for this field theory, and we will do this via the definition of the T µν tensor: ( T µν = g µν L L ) + 2 (2.20) g µν with L = F µν (A ν,µ A µ,ν ) 2 F σρ g ασ g βρ F αβ. (2.2) The most important term is the derivative w.r.t. g µν, appearing in the seond term above we an simplify life by rewriting one set of dummy indies so as to get a g µν in between the two field strengths. Remember that we ould do this on either of the two metris, so we will pik up an overall fator of 2 again using Minkowski, the usual gradient is a ovariant tensor, while the fourpotential A µ is naturally ontravariant, so it is A µ that is represented by ( V, Ax, A y, A z ) for example. Raising and lowering hanges the sign of the zero omponent only. 5 of 8

6 2.2. ENERGY-MOMENTUM TENSOR FOR E&M Leture 2 two: g µν ( ) 2 F σρ g ασ g βρ F αβ = ( ) g µν 2 F νρ g µν g βρ F µβ = F νρ g βρ F µβ. (2.22) Putting this together with the field relation between F µν and the derivatives of A µ, we have ( ) ) 2 T µν = (g µν F αβ F αβ 2 F νρ g βρ F µβ = 2 (F µβ F νβ 4 ) (2.23) gµν F αβ F αβ. Overall fators in a free Lagrange density do not matter (field equations equal zero, so there is no need to worry about onstants). For oupling to matter, however, we must put in the appropriate units. One way to do this is to onnet the stress tensor to its known form. Suppose we started from the ation S V α S V, then the energy-momentum tensor is T µν = 2 α (F µβ F νβ 4 ) gµν F αβ F αβ. (2.24) Take the zero omponent of this, T 00 = α (B B + ) 2 E E. (2.25) The usual expression for energy density is given by (Griffiths (8.32), for example): u = ) (ɛ 0 E 2 + µ0 B 2 = ɛ 0 ( E B 2) (2.26) 2 2 so evidently, we an identify T 00 = 2 α ɛ 0 2 u T 00 = u for α = ɛ (2.27) With this fator in plae, we have the final form ( T µν = 2 ɛ ( 0 E B 2) ɛ 0 E B ɛ 0 E B T ij ), (2.28) 6 of 8

7 2.2. ENERGY-MOMENTUM TENSOR FOR E&M Leture 2 with T ij the usual Maxwell stress tensor: T ij = ɛ 0 ( E i E j 2 δij E 2 ) + ɛ 0 2 ( B i B j 2 δij B 2 ). (2.29) We have the assoiation T 0j = ɛ 0 E B, the familiar momentum density from E&M. The energy density is indeed the T 00 omponent, and the spatial portion is (the negative of) Maxwell s stress tensor Units The above allowed us to set SI units for the ation via the prefator ɛ = 2 µ 0, the eletromagneti ation reads: S V = dτ g (F µν (A ν,µ A µ,ν ) 2 ) 2 µ F µν g αµ g βν F αβ, (2.30) 0 but this is not the most obvious, ertainly not the most usual, form for the ation formulation of E&M. Most ommon are gaussian units, where E and B have the same units. The basi rule taking us from SI to gaussian is: ɛ 0 4 π, µ 0 4 π (ensuring that 2 ɛ 0 µ 0 = 2 ), and B B. For example, the eletromagneti field energy in a volume in SI units is: U = ( ɛ 0 ESI 2 + ) BSI 2 dτ (2.3) 2 µ 0 using our onversion, we have U = ( ) 2 4 π E2 + 2 BSI 2 dτ µ 0 = (E 2 + B 2) dτ. 8 π (2.32) We an onvert the ation itself in its usual form, we write the ation entirely in terms of the field strength tensor F µν, so we lose the onnetion between the potential and the fields. S V = 6 π dτ F µν F µν, (2.33) where the 2 that was in the numerator from µ 0 4 π got soaked into 2 the field strength tensor beause it is quadrati, the integrand depends 7 of 8

8 2.3. CONSERVATION OF ANGULAR MOMENTUM Leture 2 on omponents like Ei E j and the 2 out front kills that, and in addition, 2 BSI i Bj SI whih beomes, upon multipliation by 2, just B i B j. In other words, the field strength tensor in gaussian units looks like (2.9) with. As a final note, you will generally see this ation written with an overall minus sign that is onvention, and leads to a ompensating sign in the stress tensor definition regardless, keep the physis in mind, and you will be safe. 2.3 Conservation of Angular Momentum Given the onservation of the energy-momentum tensor itself, we an also onstrut the angular momentum analogue define: then we have the derivative: M αβγ x β T γα x γ T βα, (2.34) M αβγ x α = T γβ T βγ = 0, (2.35) from the symmetry of the stress tensor. This gives us a set of onserved quantities in the usual way: or, in integral form: d dt M 0βγ t M 0βγ dτ = M iβγ = x i (2.36) M iβγ da i. (2.37) The left-hand side represents six independent quantities of most interest are the spatial omponents let J αβ = M 0βγ, then the densities are just as we expet: J ij = M 0ij = x i T j0 x j T i0, (2.38) so that, in terms of the spatial omponents of momentum, we have: preisely the angular momentum. J ij = x i p j x j p i, (2.39) 8 of 8