Quantum Field Theory

Perturbation Theory and S Matrix

It is not possible to obtain an analytical solution for the field equation with the field itself in the interaction term. A perturbation theory was developed to obtain approximate solutions step by step.

For example, the interaction between a charged fermion and the photon can be expressed in the form H_I = , where A is the electromagnetic vector potential and is the 4 components spinor separated into 2 parts (in Dirac representation,

and

together form a 4-components field for either the fermion or anti-fermion; while in chiral representation they stand for the left-handed and right-handed fermions called Weyl spinors). In term of the Green's function (with (y) replaced by the interaction term, replacing

, and G defined by Eq.(9a)), the function now appears on both sides in Eq.(8). The equation can be solved by an iteration procedure yielding a sum of integrals as shown in the formula below:

---------- (10)

In this form the unknown field on the left hand side is now expressed in terms of all known quantities on the right hand side. The free field solution is denoted by

₀, and the integration is over all the space-time x', x'', x''', ... Note that each of the following term is multiplied by the power of e, from e¹, to e², ... Since e²=1/137 for the electromagnetic interaction, computation on a few terms would be sufficient to obtain a result with acceptable accuracy.

Another formulism is the S-matrix. It is the transition amplitude expressed in a series as the result of the iteration procedure on the transition operator, which transforms the system from an initial state (at negative infinity time) to a final state (at positive infinity time). Derivation of the formula starts with the time evolution of an unitary operator U(t₀,t₀) = 1

In this picture the fields obey the free field equations, the interaction enters via H_I =

d³x, where

is the Hamiltonian density. The green's function and S-matrix formulations are equivalent since the transition probability amplitude

_f(t)|

_i(t)

in the former case is the same as

f|S|i

_f|S|

in each term of the expansion. The difference in appearance is due to different representation on placement of the time dependent Hamiltonian H_I. While the Green's formulation follows the Schrodinger picture with time residing in the wave function; the time-dependence part is within the S-matrix operator itself by adopting the Heisenberg picture. The difference is readily apparent by considering the source and interacting Hamiltonian to be independent of time, e.g., equals to H. By equating the t_i's to t, and performing a time-order re-arrangement, it can be shown that Eq.(10) becomes

(t) = e^-iH(t-t₀)

(t₀), while Eq.(11) turns into S = e^-iH(t-t₀).

The S-matrix elements S_fi =

f|S|i

of the expansion are taken between the initial state (i) at t₀ = -

and the final state (f) state at t_n = +

. Usually, one initial state can produce one or more final states as shown in Figure 01g, where three different initial states are taken as examples - namely, the electron positron scattering, the Compton scattering and the deep inelastic scattering. Each of this process produces many final states, but only a few have been shown just for illustration purpose. The S_fi is a complex number in general. It is called probability amplitude and is related to the probability of going from the i to f states. It has to satisfy the unitary condition, i.e.,

_f S*_fiS_fi = 1, which guarantees that probability is conserved in the process. Such relationship indicates that the matrix S_fi has an inverse, which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the second law of thermodynamics is "almost" never violated. This property is also related to the conservation of information, which caused so much trouble for Stephen Hawking.

Figure 01g S-Matrix
[view large image]

Note: For example in the electron positron scattering process, there are three possible finally states as illustrated in Figure 01g. The sum of the probabilities for each one of these has to be: S*₁₁S₁₁ + S*₂₁S₂₁ + S*₃₁S₃₁ = 1 to insure that the final state is in one of the three possibilities.

Now let us take the nucleon-pion system as an example of S-matrix application:

---------- (12)
---------- (13)

where Eq.(12) is the free field equation for the pion, and Eq.(13) is the free field equation for the nucleon (the Dirac equation). According to the derivation of Green's function, the quantization rules in terms of the field itself become :

---------- (14)

where {a,b} = ab + ba is the anticommunition expression, and the quantities on the right-hand side are the Green's functions for the pion and nucleon respectively (see Eqs.(9a) and (9b)), and also referred to as propagator. Since the interaction H_I = g_od³x the nth order term in the S-matrix expansion Eq.(11) has the explicit form:

---------- (15)

where the integration runs from -

to +

over all space-time,
each fermion spinor has dimension cm^-3/2,
the boson field's unit is erg ~ cm^-1 in natural unit (see for example vector potential),
N is the normal-order operator, which shifts all the creation operators to the left (the consequence of dropping the zero point energy),
T is the time-order operator, which re-arranges the fields so that the one associated with later time is on the left (to make sure a particle is created before its annihilation).

The T product can be expanded into the N products with pairings. The permutation has to be arranged by applying the commutative relations in Eq.(14). For example in the 2nd order S matrix :

The table below provides a graphic representation of all the terms.

If the coupling constant g₀ is a small number less than 1, the successive higher terms would be getting even smaller as the proportional constant is in the form of (g₀)ⁿ. Therefore the perturbation series can be terminated up to certain term depending on the requirement of accuracy. See more on Feynman diagrams and rules and Divergence for renormalization of the loop diagrams.

It was thought that since we cannot measure the fields directly, so we should not talk about it, while we do measure S-matrix elements, so this is what we should be mindful about. Essentially, the stringent requirements of conservation of probability, analyticity, various invariences and symmetries, ... are considered to be sufficient to determine the S matrix uniquely in a form that is in agreement with some experiments, e.g., the pion-nucleon scattering amplitude. Such methodology would bypass the tedious renormalization process at higher order involving hundreds of Feynman diagrams. This alternate method is called S-matrix Theory, which was very popular in the 1960's. It is now realized that analyzing the S-matrix alone is not sufficient, information on the quantum fields is also necessary. The followings provide just a short glance on a part of its formulation, which has considerable ramifications into the Theory of String.

The S-Matrix Theory (see its modern incarnation in Unitarity Method, and don't confuse with the S-Matrix Perturbation Theory) was a failed attempt to understand the strong interaction from a macroscopic point of view. Its part on "duality" is said to provide a base for the development of the "Theory of String", which is equally unsuccessful so far (as of 2016) without any experimental support. Anyway, not much was known about the strong interaction in the 1960's. Scattering between hadrons was the main tool to study the force. There were two kinds of such process known as the s- and t-channels.

		The s-channel corresponds to two incoming particles with 4-momenta p₁ and p₂, which combine into an intermediate (virtual) particle with real mass and finally emerge as two outgoing particles with 4-momenta p₃ and p₄ (Figure 01h,a). Mathematically, s = (p₁ + p₂)² = (p₃ + p₄)² = 4E², where we assume they all have energy E to simplify the formula. In t-channel, particle with p₁ emits virtual particle with imaginary mass, which is absorbed by p₂. They depart with p₃ and p₄ respectively (Figure 01h,b). Thus, t = (p₁ - p₃)² = (p₄ - p₂)² = 2E²(1 - cos), where is the angle between
Figure 01h Scattering Channels [view large image]	Figure 01i Regge Trajectory [view large image]	p₁ and p₃. There is an additional u-channel with the role of particles 3, 4 interchanged. Together they form an identity : s + t + u = _i=1,4 m_i².

The scattering amplitude for a simple t-channel interaction is A(s,t) ~ -g²/(t - m²) as shown by the nucleon-nucleon scattering according to the prescription associated with the Feynman diagram, where g is the coupling constant and m the mass of the intermediate particle. A similar expression is applicable for the s-channel A(t,s) ~ -g²/(s - m²). This form of scattering amplitude (for intermediate particle spin J < 1) is well behave as it vanishes for s or |t|

. As shown by QED, it has a logarithmic divergence for J = 1, and becomes nasty unrenormalizable divergence for J > 1 (e.g., for the case of graviton).

Experiments during that period uncovered a huge number of resonances (particles with very short life-time < 10^-23 sec). They have high spin quantum number and show regularity known as Regge trajectory :

(0) +

' s, where

= J whenever s = m² and

' ~ 1/(Gev)² is the Regge slope (Figure 01i).

For scalar particles at high energy the scattering amplitude can be generalized to :
A(s,t) = -

_J g_J²(s)^J/(t - m_J²) for the t-channel, and similarly
A(t,s) = -

_J g_J²(t)^J/(s - m_J²) for the s-channel.
The finite summation in these formulas yields values way above the experimental data (Figure 01j). Then it was suggested that an infinite sum may offer a more acceptable result. At about the same time, there were hints that the scattering amplitudes from the two different channels may be equal to each other, i.e., A(s,t) ~ A(t,s), which is now known as the "duality" hypothesis.

Figure 01j Hadron Scattering [view large image]

	Then a scattering amplitude (known as the Veneziano Amplitude) that satisfies all the stringent requirements of the S-matrix Theory (except the unitarity but including the duality) was discovered in 1968 :
Figure 01k Gamma Function [view large image]	Some comments on the Veneziano amplitude :

Duality between the s and t channels is obvious from the expression in terms of the Gamma Function (Figure 01k). The Regge trajectory emerges from the poles in the series expansion of the Euler beta function (Figure 01l), each one of them yields a resonance with m² = [n - (0)]/', where n is identified to the spin quantum number J. For example, in the case of ' = 1 and (0) = 2, the graviton would appear with n = J = 2, and m² = 0; while it would be a tachyon with n = J = 1 and m² = -1.
The asymptotic behavior can be considered as an average of all the zeros and poles resulting in a fictitious particle with J_eff = (t_eff), where t_eff - is off the mass-shell. Thus, the high energy behavior of this dual models can be much softer than that of any field theory.

Figure 01l Euler Beta Function [view large image]

On the other hand, the differential cross-section also falls off exponentially with s. However, despite such desirable features, it soon became obsolete as experimental data favor the Quantum Chromo-Dynamics (QCD) model for the correct theory of strong interaction.

Subsequently, the formula is found to have some peculiar properties :

In the course of studying the theory of string, it was discovered that the Regge trajectory emerges naturally from its formulation (see "Classical Theory of String"). The scattering amplitude for the closed string is the generalization of the Veneziano amplitude. It is called N-point amplitude. Figure 01m shows graphically the original 4-point Veneziano amplitude and a 6-point amplitude. The formula below is the N-point amplitude denoted by A_N .


Figure 01m N-point Amplitude [view large image]

The necessity of having to devise the N-point amplitude is related to the behavior of hadron collision at very high energies. At about 50 times the rest mass energy of the proton, the dominant feature of most events is the sheer number of particles produced. This number is called the multiplicity of the reaction as shown by Figure 01n, in which the number of the product n for the various species is plotted against the squared energy.

BTW, the intermediate particles in the S-matrix theory now include all those in the Regge trajectories. This is referred to as Reggeon exchange, which represents the exchange of all the resonances with different masses and spins. It is similar to the mediation by photon in the electromagnetic interaction. The resonances are referred to as hadron excited states in QCD.

Figure 01n Multiplicity
[view large image]

For the massless spin two intermediate particle, i.e., the graviton, the high-energy t-channel scattering amplitude was in the form : A(s,t) s²/t, which is hopelessly un-renormalizable. However, this is not so with the Veneziano amplitude. Therefore, it remains to be a potentially correct description for graviton and particles of "excessively high spin" in quantum gravity.

Then Veneziano himself discovered that self-consistency condition demands the lightest particle to be a tachyon. Application of the dual model to bosonic string requires the dimension of the theory to be 26. These problems have been addressed now in the Superstring Theory.

This strange episode in the annals of theoretical physics shows that beauty in mathematics does not guarantee a correspondence in the real world as suggested by some physicists (see for example, "A Beautiful Question", also an article about the history of "Gabriele Veneziano, Strong Nuclear Force and Beta-function").