Quantum Field Theory I

78 CHAPTER 3. INTERACTING QUANTUM FIELDS
3.1.2 Transition amplitudes
The dynamicalcontentofaquantum theoryisencoded in transitionamplitudes,
i.e. the probability amplitudes for the system to evolve from an initial state |ψ i 〉
at t i to a final state |ψ f 〉 at t f . These probability amplitudes are coefficients
of the expansion of the final state (evolved from the given initial state) in a
particular basis.
In the Schrödinger picture the initial and the final states are |ψ i,S 〉 and
U S (t f ,t i )|ψ i,S 〉 respectively, where U S (t f ,t i ) = exp{−iH(t f −t i )} is the time
evolutionoperatorin theSchrödingerpicture. The”particularbasis”isspecified
by the set of vectors |ψ f,S 〉
transition amplitude = 〈ψ f,S |U S (t f ,t i )|ψ i,S 〉
It should be perhaps stressed that, in spite of what the notation might suggest,
|ψ f,S 〉 does not define the final state (which is rather defined by |ψ i,S 〉 and the
time evolution). Actually |ψ f,S 〉 just defines what component of the final state
we are interested in.
In the Heisenberg picture, the time evolution of states is absent. Nevertheless,
the transition amplitude can be easily written in this picture as
transition amplitude = 〈ψ f,H |ψ i,H 〉
Indeed, 〈ψ f,H |ψ i,H 〉 = 〈ψ f,S |e −iHt f
e iHti |ψ i,S 〉 = 〈ψ f,S |U S (t f ,t i )|ψ i,S 〉. Potentially
confusing, especially for a novice, is the fact that formally |ψ i,H 〉 and
|ψ f,H 〉 coincide with |ψ i,S 〉 and |ψ f,S 〉 respectively. The tricky part is that in
the commonly used notation the Heisenberg picture bra and ket vectors label
the states in different ways. Heisenberg picture ket (bra) vectors, representing
the so-called the in-(out-)states, coincide with the Schrödinger picture state at
t i (t f ), rather then usual t=0 or t 0 . Note that these times refers only to the
Schrödinger picture states. Indeed, in spite of what the notation may suggest,
the Heisenberg picture in- and out-states do not change in time. The in- and
out- prefixes have nothing to do with the evolution of states (there is no such
thing in the Heisenberg picture), they are simply labelling conventions (which
have everything to do with the time evolution of the corresponding states in the
Schrödinger picture).
Why to bother with the sophisticated notation in the Heisenberg picture, if
it anyway refers to the Schrödinger picture The reason is, of course, that in
the relativistic QFT it is preferable to use a covariant formalism, in which field
operators depend on time and space-position on the same footing. It is simply
preferable to deal with operators ϕ(x) rather than ϕ(⃗x), which makes the
Heisenberg picture more appropriate for relativistic QFT.
Finally, theinteractionpicturesharestheintuitiveclaritywiththeSchrödinger
picture and the relativistic covariance with the Heisenberg picture, even if both
only to certain extent. In the interaction picture
transition amplitude = 〈ψ f,I |U (t f ,t i )|ψ i,I 〉
Thisfollowsfrom〈ψ f,S |U S (t f ,t i )|ψ i,S 〉 = 〈ψ f,I |e iH0t f
e −iH(t f−t i) e −iH0ti |ψ i,I 〉 =
〈ψ f,I |U(t f ,0)U −1 (t i ,0)|ψ i,I 〉 and from U(t f ,0)U −1 (t i ,0) = U(t f ,0)U(0,t i ) =
U(t f ,t i ).