Classical Field Theory Asaf Pe’er1 January 12, 2016 We begin by discussing various aspects of classical fields. We will cover only the bare minimum ground necessary before turning to the quantum theory, and will return to classical field theory at several later stages in the course when we need to introduce new ideas. 1. The dynamics of fields Definition: a field is a physical quantity defined at every point in space and time: (~x, t). While classical particle mechanics deals with a finite number of generalized coordinates - or degrees of freedom, qa(t) (indexed by a label a), in field theory we are interested in the dynamics of fields φa(~x, t) (1) where both a and ~x are considered as labels. Thus we are dealing with a system with an infinite number of degrees of freedom - at least one for each point ~x in space. Notice that the concept of position has been relegated from a dynamical variable in particle mechanics to a mere label in field theory. As opposed to QM, in QFT the dynamical degrees of freedom are the values of φ at every single point. As opposed to QM, where X was changed into an operator, in QFT x just stays as a label. Moreover, I stress that φ is not a wave function; although it obeys certain equations that look like the same equations in QM, it has a different inter- pretation. Example of a Field (I): The Electromagnetic Field The most familiar examples of fields from classical physics are the electric and magnetic fields, E~ (~x, t) and B~ (~x, t). Both of these fields are spatial 3-vectors. In a more sophisticated treatment of electromagnetism, we derive these two 3-vectors from a single 4-component field, Aµ(~x, t) (φ, A~), where µ = 0, 1, 2, 3 shows that this field is a vector in spacetime. ≡ 1Physics Dep., University College Cork –2– The electric and magnetic fields are given by ∂A~ E~ = ~ φ and B~ = ~ A~ (2) −∇ − ∂t ∇× which ensure that two of Maxwell’s equations, ~ B~ = 0 and dB/dt~ = ~ E~ , hold ∇ · −∇ × immediately as identities. 1.1. Dynamics of fields: Lagrangian formalism Next we are interested in knowing the dynamics of fields, namely, how they evolve. We could do that in various ways. As an example, for the electromagnetic field, we could write the equations of motion, which are Maxwell’s equations. However, we choose a more concise way: the dynamics of the field is governed by a Lagrangian which is a function of φ(~x, t), φ˙(~x, t), and φ(~x, t). In all the systems we study in this course, the Lagrangian ∇ is of the form, 3 L(t)= d x (φa,∂µφa), (3) L Z where is officially called Lagrangian density, although everyone simply calls it the L Lagrangian. The action is t2 S = dt d3x = d4x . (4) L L Zt1 Z Z Note: Recall that in particle mechanics L depends on q andq ˙, but notq ¨. In field theory we similarly restrict to Lagrangians depending on φ and φ˙, but not φ¨. In principle, there is L nothing to stop depending on φ, 2φ, 3φ, etc. However, with an eye to later Lorentz L ∇ ∇ ∇ invariance, we will only consider Lagrangians depending on φ and not higher derivatives. ∇ The equations of motion can be determined by the principle of least action. We vary the path, keeping the end points fixed and require δS = 0, 4 ∂L ∂L δS = d x δφa + δ(∂µφa) ∂φa ∂(∂µφa) (5) 4 ∂L ∂L ∂L = R d x h ∂µ δφia + ∂µ δφa ∂φa − ∂(∂µφa) ∂(∂µφa) nh i o The last term is a total derivativeR and vanishes for any δφa(~x, t) that decays at spatial infinity and obeys δφa(~x, t1) = δφa(~x, t2) = 0. Note that equation 5 is similar to the one derived for particle mechanics, only now the Lagrangian depends on both the time derivate (φ˙) and the spatial derivative ( φ) of φ; hence the use of the 4-derivative ∂µφ. Requiring ∇ δS = 0 for all such paths yields the Euler-Lagrange equations of motion for the fields φa, ∂ ∂ ∂µ L L =0. (6) ∂(∂ φ ) − ∂φ µ a a –3– Example (II): the Klein-Gordon equation Consider the Lagrangian for a real scalar field, φ(~x, t), 1 µν 1 2 2 = η ∂µφ∂νφ m φ L 2 − 2 (7) = 1 φ˙2 1 ( φ)2 1 m2φ2 2 − 2 ∇ − 2 Where recall that we use the Minkowski space-time metric, 10 0 0 µν 0 1 0 0 η = ηµν = − (8) 0 0 1 0 − 0 0 0 1 − We can compare the Lagrangian in Equation 7 to the usual expression for the Lagrangian, L = T V , and identify the kinetic energy of the field as − 1 T = d3x φ˙2, (9) 2 Z and the potential energy of the field as 1 1 V = d3x ( φ)2 + m2φ2 . (10) 2 ∇ 2 Z The first term in Equation 10 is called the gradient energy, while the phrase “potential energy”, or just “potential”, is usually reserved for the last term. This also provides some “intuitive” explanation for the choice of sign in ηµν, which is opposite to that used in GR: using the sign convention in Equation 8, the kinetic term in Equation 9 is positive. In order to determine the Equation of motion associated with the Lagrangian in Equa- tion 7, we compute: ∂ ∂ L = m2φ and L = ∂µφ (φ,˙ φ). (11) ∂φ − ∂(∂µφ) ≡ −∇ The resulting Euler-Lagrange equation is thus φ¨ 2φ + m2φ =0, (12) −∇ which can be written in a relativistic form as µ 2 ∂µ∂ φ + m φ =0. (13) –4– Equation 13 is known as Klein-Gordon Equation (which was first written by Schr¨odinger). Note though that the field φ has got nothing to do with the wave function ψ in quantum mechanics. The Laplacian in Minkowski space is sometimes denoted by ✷. In this notation, the Klein-Gordon Equation read ✷φ + m2φ = 0. An obvious generalization of the Klein-Gordon equation comes from considering the Lagrangian with arbitrary potential V (φ), 1 µ µ ∂V = ∂µφ∂ φ V (φ) ∂µ∂ φ + =0. (14) L 2 − ⇒ ∂φ Example (III): first order Lagrangians We could also consider a Lagrangian that is linear in time derivatives, rather than quadratic. Take a complex scalar field ψ whose dynamics is defined by the real Lagrangian, i = ψ⋆ψ˙ ψ˙ ⋆ψ ψ⋆ ψ mψ⋆ψ. (15) L 2 − −∇ ·∇ − The equation of motion can be determined by treating ψ and ψ⋆ as independent objects, so that ∂ i ∂ i ∂ L = ψ˙ mψ and L = ψ and L = ψ. (16) ∂ψ⋆ 2 − ∂ψ˙ ⋆ −2 ∂ ψ⋆ −∇ ∇ This gives the equation of motion, ∂ψ i = 2ψ + mψ. (17) ∂t −∇ Equation 17 looks very much like the Schr¨odinger equation - only it is not !. Or, at least, the interpretation of this equation is very different: the field ψ is a classical field with none of the probability interpretation of the wavefunction. We will return to this point later on. The initial data required on a Cauchy surface 1 differs for the two examples above. When φ˙2, both φ and φ˙ must be specified to determine the future evolution; however L ∼ when ψ⋆ψ, only ψ and ψ⋆ are needed. L∼ 1Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future (and the past) uniquely. –5– Example (IV): Maxwell’s Equations Maxwell’s equations in vacuum can be derived from the Lagrangian, 1 µ ν 1 µ 2 = (∂µAν)(∂ A )+ (∂µA ) (18) L −2 2 Note the following: (I) the minus sign in front of the first term ensures that the kinetic terms (with time derivatives) for Ai are positive, using Minkowski space-time metric (Equation 8), and thus 1 A˙ 2. (II) The Lagrangian in Equation 18 has no kinetic term A˙ 2 for A , since L∼ 2 i 0 0 the two terms cancel. We can now compute ∂ µ ν ρ µν L = ∂ A +(∂ρA )η , (19) ∂(∂µAν) − from which we obtain ∂ 2 ν ν ρ µ ν ν µ µν ∂µ L = ∂ A + ∂ (∂ρA )= ∂µ(∂ A ∂ A ) ∂µF (20) ∂(∂ A ) − − − ≡− µ ν where the electromagnetic field strength tensor is defined by Fµν ∂µAν ∂νAµ. ≡ − Since is independent on the fields (only on their derivatives), Lagrange’s equation of L µν motion becomes ∂µF = 0. This is equivalent to the remaining two Maxwell’s equations in vacuum, ~ E~ = 0, and ∂E/∂t~ = ~ B~ .2 Using the notation of the field strength, we may ∇· ∇× rewrite the Maxwell Lagrangian (up to an integration by parts) in the compact form 1 µν = FµνF . (21) L −4 1.2. Locality In each of the examples above, the Lagrangian is local. This means that there are no terms in the Lagrangian coupling φ(~x, t) directly to φ(~y, t) with ~x = ~y. For example, there 6 are no terms that look like L = d3xd3yφ(~x)φ(~y). (22) Z A priori, there is no reason for this. After all, ~x is merely a label, and we are quite happy to couple other labels together (for example, the term ∂3A0∂0A3 in the Maxwell Lagrangian 2For further discussion, please compare with the GR notes on special relativity. –6– couples the µ = 0 field to the µ = 3 field). But the closest we get for the ~x label is a coupling between φ(~x) and φ(~x + δ~x) through the gradient term( φ)2. This property of locality is, ∇ as far as we know, a key feature of all theories of Nature.