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I-6: the S-Matrix and Time-Ordered Products

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I-6: the S-Matrix and Time-Ordered Products 2012 Matthew Schwartz I-6: The S-matrix and time-ordered products 1 Introduction As discussed in the previous lecture, scattering experiments have been a fruitful and efficient way to determine the particles which exist in nature and how they interact. In a typical collider experiment, two particles, generally in approximate momentum eigenstates at t = are col- lided with each other and we measure the probability of finding particular outgoing−∞ momentum eigenstates at t = + . All of the interesting interacting physics is encoded in how often given initial states produce∞ given final states, that is, in the S-matrix. The working assumption in scattering calculations is that all of the interactions happen in some finite time T<t<T. This is certainly true in real collider scattering experiments. But more importantly,− it lets us make the problem well-defined; if there were always interactions, it would not be possible to set up our initial states at t = or find the desired final states at t = + . Without interactions at asymptotic times, the states−∞ we scatter can be defined as on-shell single∞ particle states of given momenta, known as asymptotic states. In this lecture, we derive an expression for the S-matrix using only that the system is free at asymptotic times. In the next lecture we will work out the Feynman rules, which make it easy to perform a perturbation expansion for the interacting theory. The LSZ (Lehmann-Symanzik-Zimmermann) reduction formula relates S-matrix ele- ments f S i for n asymptotic momentum eigenstates to an expression involving the quantum fields φh( x|) : | i 4 − ip1 x 1 2 4 ipn x n 2 f S i = i d x1 e ( + m ) i d xn e ( + m ) (1) h | | i Z Z Ω T φ( x ) φ( x ) φ( x ) φ( xn) Ω (2) ×h | { 1 2 3 }| i with the i in the exponent applying for initial states and the +i for final states. In this for- mula, T − refers to a time-ordered product, to be defined below, and Ω is the ground state or vacuum{} of the interacting theory, which in general may be different from| i the vacuum in a free theory. The time-ordered correlation function in this formula can be very complicated and encodes a tremendous amount of information besides S-matrix elements. The factors of + m2 project onto the S-matrix: + m2 becomes p2 + m2 in Fourier space which vanishes for the asymp- totic states. These factors will therefore− remove all terms in the time-ordered product except 1 those with poles of the form p2 − m 2 , corresponding to propagators of on-shell particles. Only the terms with poles for each factor of p2 m2 will survive, and the S-matrix is given by the residue of these poles. Thus the physical content− of the LSZ formula is that the S-matrix projects out single-particle asymptotic states from time-ordered product of fields. 2 The LSZ reduction formula [The following proof follows Srednicki. You can find different versions in various textbooks.] In the last lecture, we derived a formula for the differential cross section for 2 n scattering of asymptotic states → 1 2 dσ = d ΠLIPS (3) Q (2 E )(2 E ) Qv v |M| 1 2 | 1 − 2 | where d ΠLIPS is the Lorentz-invariant phase space, and , which is shorthand for f i , is the S-matrix element with an overall momentum-conservingM δ-function factored out: h |M| i f S 1 i = i(2 π) 4 δ4(Σ p) (4) h | − | i M 1 2 Section 2 The state i is the initial state, at t = and f is the final state, at t = + . More precisely, | i † −∞ h | ∞ using the operators a p( t) which create particles with momentum p at time t, these states are i =√ 2 ω √2 ω a † ( ) a † ( ) Ω (5) | i 1 2 p1 −∞ p2 −∞ | i where Ω is the ground state, with no particles, and | i † † f =√ 2 ω √2 ωn a ( ) a ( ) Ω (6) | i 3 p3 ∞ pn ∞ | i We are generally interested in the case where some scattering actually happens, so let us assume 1 f i in which case the does not contribute. Then the S matrix is | i | i n/2 † † f S i = 2 √ω ω ω ωn Ω a p3 ( ) a p ( ) a ) a ( ) Ω (7) h | | i 1 2 3 | ∞ n ∞ p1 − ∞ p2 −∞ | This expression is not terribly useful as is. We would like to relate it to something we can com- pute with our Lorentz invariant quantum fields φ( x) . Recall that we had defined the fields as a sum over creation and annihilation operators 3 d p 1 − ipx † ipx φ( x) = φ( xQ , t) = a p( t) e + a ( t) e (8) Z (2 π) 3 2 ω p p p 2 2 Q where ωp = Qp + m . We also start to use the notation φ( x) = φ( x , t) as well, for simplicity. These are Heisenberg-picturep operators which create states at some particular time. However, the creation and annihilation operators at time t are in general different from those at some other time t ′. An interacting Hamiltonian will rotate the basis of creation and annihilation oper- ators, which encodes all the interesting dynamics. For example, if H is time-independent, iH ( t − t 0 ) − iH ( t − t 0 ) iH ( t − t0 ) − iH ( t − t0 ) a p( t) = e a p( t0) e just like φ( x) = e φ( Qx , t0) e where t0 is some arbitrary reference time where we’ve matched the interacting fields onto the free fields. We won’t need to use anything at all in this section about a p( t) and φ( xQ , t) except that these opera- Q iQ x p tors have some ability to annihilate fields at asymptotic times: Ω φ( Qx , t = ) p = Ce for some constant C, as was shown for free fields in Lecture I-2. h | ±∞ | i The key to proving LSZ is the algebraic relation 4 ipx 2 i d xe ( + m ) φ( x) = 2 ωp[ a p( ) a p( )] (9) Z ∞ − − ∞ p µ Where p = ( ωp,Q p ) . To derive this, we only need to assume that all the interesting dynamics happens in some finite time interval, T<t<T so that the theory is free at t = ; no assumption about the form of the interactions− during that time is necessary. ±∞ To prove Eq. (9), we will obviously have to be careful about boundary conditions at t = . ±∞ However, we can safely assume that the fields die off at xQ = allowing us to integrate by ±∞ parts in Qx . Then, 2 4 ipx 2 4 ipx 2 Q 2 i d xe ( + m ) φ( x) =i d xe ∂t ∂x + m φ( x) Z Z − 4 ipx 2 2 2 =i d xe ( ∂t + Qp + m ) φ( x) Z 4 ipx 2 2 =i d xe ( ∂t + ωp) φ( x) Z Note this is true for any kind of φ( x) , whether classical field, or operator. Also, ipx ipx ipx 2 ∂t[ e ( i∂t + ωp) φ( x)] =[ iωp e ( i∂t + ωp) + e ( i∂t + ωp∂t)] φ( x) ipx 2 2 =ie ( ∂t + ωp) φ( x) which holds independent of boundary conditions. So, 4 ipx 2 4 ipx i d xe ( + m ) φ( x) = d x∂t[ e ( i∂t + ωp) φ( x)] (10) Z Z Q iω p t 3 − iQ p x = dt∂t e d xe ( i∂t + ωp) φ( x) (11) Z Z The LSZ reduction formula 3 Again, this is true for whatever kind of crazy interaction field φ( x) might be. This integrand is a total derivative in time, so it only depends on the fields at the boundary † t = . By construction, our a p( t) and a ( t) operators are time-independent at late and early ±∞ p times. For the particular case of φ( x) being a quantum field, Eq. (8), we can do the xQ integral. d3 k Q Q Q 3 − iQ p x 3 − i p x 1 − ikx d x e ( i∂t + ωp) φ( x) = d x e ( i∂t + ωp) ak ( t) e + Z Z Z (2 π) 3 √2 ω k † ikx ak ( t) e (12) d3 k ω ω ω ω Q Q Q 3 k + p − ikx − iQ p x k + p † ikx − i p x = 3 d x ak ( t) e e + − ak ( t) e e (13) Z (2 π) Z √2 ωk √2 ωk Here we used ∂tak ( t) = 0, which is not true in general, but true at t = where the fields are ±∞ 3 Q Q free, which is the only region relevant to Eq. (11). The xQ integral gives a δ ( p k ) in the first − 3 Q term and a δ ( Qp + k ) in the second term. Either way, it forces ωk = ωp and so we get Q 3 − iQ p x − iω p t d xe ( i∂t + ωp) φ( x) = 2 ωp a p( t) e (14) Z Thus, p 4 ipx 2 iω p t − iω p t i d xe ( + m ) φ( x) = dt∂t[( e ) 2 ωp a p( t) e ] Z Z p = 2 ωp[ a p( ) a p( ) ∞ − − ∞ which is what we wanted. Similarly (by takingp the Hermitian conjugate) † † 4 − ipx 2 2 ωp a ( ) a ) = i d xe ( + m ) φ( x) (15) p ∞ − p − ∞ − Z p Now we’re almost done. We wanted to compute n † † f S i =√ 2 ω ωn Ω a p3 ( ) a p ( ) a ) a ( ) Ω (16) h | | i 1 | ∞ n ∞ p1 − ∞ p2 −∞ | † and we have an expression for a p( ) a p( ) .
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