S-matrix and transition amplitudes
TRANSITION AMPLITUDE AND |in> AND |out> STATES
In high-energy collisions, for instance at accelerator-based experiments, two particles
are approaching each other, acting (asymptotically) like free particles. Then they collide
and interact. The result of this interaction is a bunch of particles leaving the
interaction region and, again, when they are far enough from each other, act like free
particles. Measurements of such processes are based on cross sections, hence, from the
theory point of view, the aim is to calculate them. To do so, it suffices for the moment
to state that cross sections are proportional to the probability for the corresponding
process to occur, i.e. to the square of the quantum mechanical transition amplitude. An
amplitude for a process that transforms an incoming state ``in'' into an outgoing state
``out'' is given by
Experimentally, in most cases, the state |in> is prepared and the state |out> is measured.
From a theory point of view, assuming that the two states may be represented by asymptotically
free particles which do not experience any interaction, the |in>-state is prepared in the
(infinite) past and the out state is measured in the (infinite) future. Hence
Such states obey completeness relations, since they have to span the full Fock space
of all possible states,
To understand this a bit better, think of an harmonic oscillator, the prime example of
quantum mechanics. There, the α and β would just represent occupation numbers.
However, here it is important to note that these relations are valid for equal times T
only. For large times then, the states are simple to construct; since, by construction,
they are asymptotically free, they are just plane waves, eventually with some fixed
vector or so multiplying them.
CONSTRUCTING THE STATES
Labelling one-particle states with their four momentum p (and ignoring spin for the moment),
they are normalised such that
The various factors won't be explained in detail here, it should suffice to say that in
the first equation, the δ function incorporates the relativistic energy-momentum
relation and that the θ function constrains the states to those with positive energies
only. The factor of twice the energy in the second equation stems from using the
energy-momentum relation on each state; it is just one of the properties of the δ
function. The asymptotically free, non-interacting states can then be constructed as products
of such one-particle states.
Using a sketch of quantum theory, the statement above trasnlates into the following:
The individual, non-interacting quanta can be created or annihilated by corresponding
creation or annihilation operators, indexed with a momentum,
and the many-particle states can be produced by repeatedly applying creation operators
on the vacuum
The vacuum state |0> in these equations is defined to be Lorentz invariant, non-degenerate,
and translationally invariant, and it is assumed that it has zero energy. It also
is annihilated through annihilation operators, i.e.
Also, a particle number operator can be defined, etc.. The fields of course are operators
themselves, in general they can be written as a linear combination of annihilation and
destruction operators. The construction sketched here is often called "second quantisation"
and it is the subject of corresponding theory lectures.
DEFINITION OF THE S-MATRIX
From quantum mechanics the concept of unitary transformations, connecting states is known.
Such a transformation can also be defined which translates the |in> into |out> states and
vice versa. The corresponding operator is known as the S-matrix, defined through
which obviously contains all information concerning the interaction that translates |in> into
|out> states. The fact that the S-matrix is a unitary operator can be easily seen from the
completeness relation above. In particular
The meaning is clear: Since the S-matrix is unitarity, it
conserves probability; stated in other words : no |in> state can disappear.
However, most of the time, nothing really happens in
scattering experiments, the particles just "miss" each other. Therefore, one may write
where now the T-matrix contains all intersting information. Sandwiching the S-matrix in this
form between states implies that
Taking into account that Lorentz-invariance is one of the sacred principles of modern
field theory, one may rewrite the transition amplitude in terms of the T-matrix,
supplemented with four-momentum conservation as
where the momenta in the &delta function are the summed momenta of the initial and final
state α and β.
TRANSITION PROBABILITY
Then, the probability for a process with non-identical initial and final state to happen is
proportional to the transition amplitude squared,
where the time labels are suppressed from now on. It should be noted that this transition
probability is over all of space-time; thus, in a next step, it'll be normalised to unit
elements of space-time. This, however, is easier than it seems. Just consider the square
of the δ-function,
Examining the integral definition of the δ-function in a box, it becomes obvious that
is just the volume V of space-time. Hence,
is the transition rate per unit space-time volume.