Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by addition and deletion. They are intended for use by class participants only, and are not for circulation. November 2, 2011 Week 5-6: Lectures 9 11 The Charged Scalar Field The charged scalar field is the quantized versiion of the classical complex scalar field. in alternate coordinates ( creation and annihilation operators) labelled by wave numbers, or equivalently, momenta. We recall that the complex scalar field can always be written as a sum of two real scalars, Φ = φ 1 + iφ 2, so the quantization of the complex field closely follows the procedure for the real field. Of special interest are Lagrange densities that are form invariant under phase transformation, of the general form L = Φ (x) Φ(x) m 2 Φ(x) 2 V ( Φ 2 ), for some function V ( Φ 2 ), where the simplest choice is proportional to ( Φ 2 ) 2, a so-called Φ to the fourth or just Φ 4 interaction. We start by identifying the canonical momenta of the field and its hermitian conjugate, Π(x) = L 0 Φ = 0Φ (x), Π L (x) = 0 Φ = 0Φ(x). The ETCRs are Φ( x, x 0 ), Π( y, x 0 ) ] = i hδ 3 ( x y) Φ ( x, x 0 ), Π ( y, x 0 ) ] = i hδ 3 ( x y) Φ ( x, x 0 ), Π( y, x 0 ) ] = Φ( x, x 0 ), Π ( y, x 0 ) ] =0, 19
which is like two copies of the real scalar case. This two-to-one correspondence is maintained in the expansion of the fields in plane waves, and leads to an interpretation in terms of particles and antiparticles. In the expansion in terms of alternate coordinates, the coefficients of complex conjugate solutions must be treated as independent, Φ(x) = d 3 k ( a+ ( (2π) 3/2 2ω(k) k, x 0 )e i k x + a ( k, x 0 )e ) i k x The alternate coordinates obey equal-time canonical commutation relations, which follow from those of the fields, a± ( k, x 0 ),a ±( k,x 0 ) ] = 2ω(k) δ 2 ( k k ) a± ( k, x 0 ),a ( k, x 0 ) ] = 0. If we follow the same procedure as for the free fields, we find a Hamiltonian, and conserved Noether charge (H, Q] = 0) : H : = 1 2 d 3 k a + ( k)a + ( k)+a ( k)a ( k) ], which describe two decoupled systems of plane-wave excitations. We have normal ordered to eliminate zero-point energies, as discussed before for the real scalar field. The time-dependence of the as and a s can be solved from the general relation (in the Heisenberg picture of time development see later for a review) O(x 0 ),H]=i hȯ(x 0), and as in the free case a( k, x 0 ) = a( k) e iω( k)x 0 a ( k, x 0 ) = a ( k) e iω( k),x 0 so that time dependence cancels in each term in the Hamiltonian, as indicated above, and the expansion of the field itself is Φ(x) = d 3 k ( a+ ( (2π) 3/2 2ω(k) k)e i k x + a ( k)e ) i k x, where as for the real field, we define k µ =(ω( k), k), with k 2 = m 2. 20
For the complex field, we can also construct the conserved charge for the Noether current associated with phase invariance, Q = d 3 k 2ω(k) a + ( k)a + ( k)+a ( k)a ( k) ]. States are then built out of the vacuum in the same way as for the real scalar field, but because there are now two creation operators, we have two kinds of states, particle states, { { k (+) i }, and antiparticle states, { { k ( ) i }. An state with a definite number of particles is some combination of particles and antiparticles, { k (+) i }{ k ( ) j } = i a +( k i ) j a ( k j ) 0. The energy of such a state is just the sum of the energies of all these particles: : H : { k (+) i }{ k ( ) j } = i ω i ( k i )+ j ω j ( k j ) { k (+) i }{ k ( ) j }, while the Noether charge of such a state is just the difference in their numbers Q { k (+) i }{ k ( ) j } = ( i ) ( j ) { k (+) i }{ k ( ) j }. 21
Particles States A general single-particle state (in Hilbert state H 1 ) is χ = d 3 k 2ω(k) f χ( k) a ( k) 0. Here we are using the real scalar, but particle and antiparticle states for the complex scalar field are constructed the same way. We can define position eigenstates as x, x 0 = d 3 k 2ω(k)] 1/2 ei k x k x, x 0 x,x 0 = δ 3 ( x x ) and in these terms define normalizable single-particle wave functions, ψ χ ( x, x 0 ) = x, x 0 χ. The property of Bose symmetry follows from the equal-time CRs. Bose symmetry is a property of multiparticle states. Consider a general twoparticle state, χ 2 = d 3 k1 d 3 k2 2ω(k 1 )2ω(k 2 ) f χ( k 1, k 2 ) a ( k 1 ) a ( k 2 ). We can use the ETCRs of the a s and a simple change of variables to verify that the function f χ ( k 1, k 2 ) gives the same 2-particle state as the function in which its arguments are exchanged: f χ ( k 2, k 1 ). Notice that in general this is not the same function. But because the states are the same, we may as well start with the explicitly symmetrized form for f χ, f χ (sym) ( k 1, k 2 ) (1/2) f χ ( k 1, k 2 )+f χ ( k 2, k 1 )]. Put equivalently, the commutation relations ensure that any antisymmetric parts of the wave function f χ do not contribute to the state. These considerations are easy to extend to any number of particles. 22
Green Functions Classical Green Functions A Green function for scalar fields is the solution to the equation ( µ µ + m 2) G(x x )=δ 4 (x x ) (5) This equation is most easily solved after a Fourier transform, where it becomes algebraic, ( k 2 m 2) G(k) = 1 tildeg(k) = 1 k 2 m 2. We set aside for a moment the question of how to treat k 2 = m 2, and recall how such a Green function is used. The method is one that should be familiar from classical electromagnetism. Thus, we can begin with a Lagrange density where a free field is coupled to a source, J(x), L = 1 2 ( µ φ) 2 m 2 φ 2] φ J. The source is regarded an arbitrary function, regarded as fixed, and we are to calculate the field in the presence of the source. The equation of motion is thus ( µ µ + m 2) φ(x) = J(x), which is precisely solved by the use of our Green function, φ(x) = d 4 yg(x y) J(y). (6) This is what Green functions are designed to do solve linear equations with arbitrary sources. This is fine for the free field, where the solution is linear in the source, another consequence of the superposition principle for free fields. But its use is more general, and we can apply the Green function to an interacting field theory to find, if not an exact solution, then at least an approximation scheme, called perturbation theory. 23
As an example, we consider the simplest interacting potential, that of φ 3 theory, L = 1 2 ( µ φ) 2 m 2 φ 2] (g/6)φ 3 φ J. with g a small number. Now in four dimensions, g is has dimensions of mass, or inverse length in natural units. We will come back to this issue shortly. Our equation of motion is now ( µ µ + m 2) φ(x) = J(x) g 2 φ2 (x), (7) which we solve iteratively by expanding our field as a power series in g, φ(x) =φ 0 (x)+gφ 1 (x)+g 2 φ 2 (x)+.... (8) Inserting (8) in (7), we demand that the coefficients of g n be equal on both sides of the resulting expression. At lowest order, we rederive φ 0 in Eq. (6), while at order g, we derive ( φ 1 (x) = d 4 yg(x y) g ) 2 φ2 0(y) = g d 4 yg(x y) d 4 y G(y y ) J(y ) d 4 y G(y y ) J(y ). 2 In momentum space, that is, on taking the Fourier transform, φ(q) d 4 xe iq x φ(x) the expressions for φ 0 and φ 1 become φ 0 (q) = = i q 2 m J(q) 2 i d 4 k q 2 m 2 (2π) 4 ij(k) ij(q k) k 2 m 2 (q k) 2 m, 2 which brings us back to the question of how to treat the vanishing of the denominator. 24
Green Functions for Quantum Scalar Fields We start by considering two-field vacuum matrix elements for the free scalar field. These are easy to evaluate, and we find, quite generally d 3 k 0 φ(y)φ(x) 0 = e i k (y x), (2π) 3 2ω(k) which we readily find by inserting a complete set of states and recognizing that only the single-particle states contribute. Now we note that there is a direct physical interpretation of this matrix element when y 0 >x 0. It represents a causal process: vacuum one particle state vacuum, where the action of the field operator mediates the changes in the state. First, the particle is created by the field, then (later) it is absorbed by the field. To develop this concept of causal ordering, we introduce the timeordering operation, denoted by two fields by T (φ(y)φ(x)) = φ(y)φ(x) θ(y 0 x 0 )+φ(x)φ(y) θ(x 0 y 0 ) Using this definition and the relation (d/dx)δ(±x x )=±δ(±x x ), we use the ETCRs to verify that the vacuum expectation value of the time-ordered product of fields satisfies Eq. (5), and thus really is a Green function in the usual sense. This result does not assume that this is the free field, only that π = φ. The causal two-point Green function is then defined by 0 T (φ(y)φ(x)) 0 = θ(y 0 x 0 d 3 k ) (2π) 3 2ω(k) +θ(x 0 y 0 d 3 k ) (2π) 3 2ω(k) i F (x y) In the vacuum expectation value of the time-ordered product, singleparticle states are always created by the field at the earlier position. In the last line, we define a new function, F, which is often called the Feynman, or Feynman-Stueckelberg, or causal propagator. 25
It is often convenient to express F as a four-dimensional integral in invariant form. F (x y) = i d 4 k (2π) 4 e k (x y) k 2 m 2 + iɛ We can verify that this is the same function as the one defined above by doing the k 0 integral first. We close the contour either in the upper half-plane or lower half-plane depending on the sign of x 0. That is, we identify poles: k 0 = ±ω(k) ɛ; LHP for x 0 >y 0, UHP for x 0 <y 0, and close in the half-plane where the exponential decreases for large Im(k 0 ). The conclusion of this discussion is that by defining our Green function as the causal propagator, we have found a way to define the singularities of the Green function in momentum space. We readily generalize the two-point Green function to n-points, ( n ) G n (x n... x 1 )= 0 T φ(x i ) 0, (9) i=1 where the time ordering operator acts on any number of fields by moving them into an order where their times increase from right to left (the opposite of reading in English). We will find that an enormous amount of information is contained in these Green functions, or correlation functions as they often called. To see how, we next turn to interacting theories and the role of scattering experiments. 26
Interacting Fields and Scattering Effects of interactions: H d 3 kh(k) With interactions like φ 3 or φ 4 in the Lagrange density, equations of motion become nonlinear and we loose the superposition principle. We ve already seen that in our example from classical φ 3 theory, because the perturbative solution at order g is already quadratic in the source J(x). This means that if we have two sources, J 1 and J 2, or example, the field φ is no longer the sum of the fields produced by the two sources separately. There is no exact solution in general. Still, expect states to be labelled by conserved charges, as p µ, {e i } Spectral Assumptions (convenient: not always true! but exceptions can be treated once the formalism is established.) (i) Unique ground state (ii) Continuum of single-particle states p µ, with p 2 = m 2 (iii) All additional states have p 2 4m 2 (no bound states (!)) (iv) 0 φ(x) 0 =0 Scattering states: Heisenberg in states: { p i }; in : prepared in distant past Heisenberg out states: { p i }; out : observed in distant future Single particle case: { p i }; in = { p i }; out 27
Completeness: we assume that any state will be simple if we look far enough in the past, or far enough into the future. 1 = { p i }; in { p i }; in = { p i }; out { p i }; out For interacting theories, states that describe systems that are simple far in the past are not simple far in the future. Rather, it is a superposition of states that are simple far in the future. Generally, these states will be distinguished from each other by the numbers and momenta of particles emerging from a collision. Of course, the total momentum will be conserved. The S-matrix the amplitude for a system that was simple far in the past to evolve into a system that is simple far in the future. S βα = β; out α; in The reduction formalism, which we will study next, will relate the S matrix to Green functions. 28