In quantum field theory, the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a method to calculate S-matrix elements (the scattering amplitudes) from the time-ordered correlation functions of a quantum field theory.
It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities.
It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.
[1] Although the LSZ reduction formula cannot handle bound states, massless particles and topological solitons, it can be generalized to cover bound states, by use of composite fields which are often nonlocal.
Furthermore, the method, or variants thereof, have turned out to be also fruitful in other fields of theoretical physics.
describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta {p}.
, making the assumption that in the far away past interaction described by the current j0 is negligible, as particles are far from each other.
is a solution of the homogeneous equation associated with the equation of motion: and hence is a free field which describes an incoming unperturbed wave, while the last term of the solution gives the perturbation of the wave due to interaction.
It is a free scalar field so it can be expanded in plane waves: where: The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form: where: The Fourier coefficients satisfy the algebra of creation and annihilation operators: and they can be used to build in states in the usual way: The relation between the interacting field and the in field is not very simple to use, and the presence of the retarded Green's function tempts us to write something like: implicitly making the assumption that all interactions become negligible when particles are far away from each other.
The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states
With these pieces one can state a correct and useful but very weak asymptotic relation: The second member is indeed independent of time as can be shown by differentiating and remembering that both
The out state can contain anything from the vacuum to an undefined number of particles, whose momenta are summarized by the index β.
The in state contains at least a particle of momentum p, and possibly many others, whose momenta are summarized by the index α.
With the assumption that no particle with momentum p is present in the out state, that is, we are ignoring forward scattering, we can write: because
Expressing the construction operators in terms of in and out fields, we have: Now we can use the asymptotic condition to write: Then we notice that the field φ(x) can be brought inside the time-ordered product, since it appears on the right when x0 → −∞ and on the left when x0 → ∞: In the following, x dependence in the time-ordered product is what matters, so we set: It can be shown by explicitly carrying out the time integration that:[note 2] so that, by explicit time derivation, we have: By its definition we see that fp (x) is a solution of the Klein–Gordon equation, which can be written as: Substituting into the expression for
and integrating by parts, we arrive at: That is: Starting from this result, and following the same path another particle can be extracted from the in state, leading to the insertion of another field in the time-ordered product.
A very similar routine can extract particles from the out state, and the two can be iterated to get vacuum both on right and on left of the time-ordered product, leading to the general formula: Which is the LSZ reduction formula for Klein–Gordon scalars.
It gains a much better looking aspect if it is written using the Fourier transform of the correlation function: Using the inverse transform to substitute in the LSZ reduction formula, with some effort, the following result can be obtained: Leaving aside normalization factors, this formula asserts that S-matrix elements are the residues of the poles that arise in the Fourier transform of the correlation functions as four-momenta are put on-shell.
Recall that solutions to the quantized free-field Dirac equation may be written as where the metric signature is mostly plus,
of non-interacting particles approaching an interaction region at the origin, where scattering occurs, followed by an out state
The probability amplitude for this process is given by where no extra time-ordered product of field operators has been inserted, for simplicity.
Now recall that in the free theory, the b-type particle operators can be written in terms of the field using the inverse relation where
Rewriting the limits in terms of the integral of a time derivative, we have where the row vector of matrix elements of the barred Dirac field is written as
, substituting it into the first term in the integral, and performing an integration by parts, yields Switching to Dirac index notation (with sums over repeated indices) allows for a neater expression, in which the quantity in square brackets is to be regarded as a differential operator: Consider next the matrix element appearing in the integral.
, we can replace the annihilation operators with in fields using the adjoint of the inverse relation.
Applying the asymptotic relation, we find Note that a time-ordering symbol has appeared, since the first term requires
The reason of the normalization factor Z in the definition of in and out fields can be understood by taking that relation between the vacuum and a single particle state
with four-moment on-shell: Remembering that both φ and φin are scalar fields with their Lorentz transform according to: where Pμ is the four-momentum operator, we can write: Applying the Klein–Gordon operator ∂2 + m2 on both sides, remembering that the four-moment p is on-shell and that Δret is the Green's function of the operator, we obtain: So we arrive to the relation: which accounts for the need of the factor Z.
On the other hand, the interacting field can also connect many-particle states to the vacuum, thanks to interaction, so the expectation values on the two sides of the last equation are different, and need a normalization factor in between.
The right hand side can be computed explicitly, by expanding the in field in creation and annihilation operators: Using the commutation relation between ain and