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Wave Equations

Second Quantization and Feynman Diagram

See the 2021 version.

Second quantization can be shown in a more concise way without taking care of all kinds of components by using the the scalar field as an example. The general solution for the Klein-Gordon equation of Eq.(18) is just the expansion :
----------(54)
where k is the 4-momentum (kx, ky, kz, iE), and x is the spacetime (x, y, z, it). Second quantization also adheres to the uncertainty principle as prescribed by Eqs.(1), and (2). Instead of the position of the particle, one of the variable is now the wave function , while the "canonical momentum" is just = in this case. Then the rule for second quantization becomes :
- = i ---------- (55)
leading to the commutative relations after integrating over all space :
---------- (56)
where (ck)ij = (ck*)ji, when ck is in the form of a matrix such as :
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where the column matrix represents the state of the system with each matrix element stands for vacuum (0 # of particle state), 1, 2, ... particles and so on. The 0 and 1 denote unoccupied and occupied respectively. The example above shows the 2 particles state is occupied.

If a number operator Nk = ckck is defined such that it operates on the state vector |nk to generate:
Nk|nk = nk|nk
where nk is the number of particles in the k state; it follows that
ck|nk = (nk+1)1/2|nk+1
ck|nk = nk1/2|nk-1
Thus ck* increases the number of particles in the k state by 1, while ck reduces the number of particles in the k state by 1. They are called creation and annihilation operator respectively. The complete set of eigenvectors is given by:
---------- (57)
for all values of kl and nl. They form an abstract space called the Fock space with all the eigenvectors orthogonal (perpendicular) to each other and the norm (length) equal to 1.

In particular, the vacuum state is:
---------- (58)
which corresponds to no particle in any state - the vacuum.

Since ckck'|0 = |1(k),1(k') = ck'ck|0 = |1(k'),1(k) ---------- (59)
by virtue of the commutative relation in Eq.(56). In case when k = k', Eq.(59) becomes :
ckck|0 = |2(k) ---------- (60)
which suggests that two or more scalar particles can occupy the same state. This turns out to be the property of all bosons having integral spin and they all obey the Bose-Einstein statistics.

For spin 1/2 particles, the Dirac spinor can be expanded as :
---------- (61)
where the superscript r denotes the 4 kinds of Dirac spinors as shown in Eq.(31). The Pauli exclusion principle imposes a restriction, which changes the commutative relations to anti-commutative relations :
---------- (62)
where {A,B} = AB + BA, and (b)ij = (b*)ji, the b's assume the same role as the c's except that the state vector can have only 0 or 1 particle for a given state (p, r). For example, exchange of states for two particles produces a minus sign in the state vector :
bpbp'|0 = |1(p),1(p') = -bp'bp|0 = -|1(p'),1(p) ---------- (63)
by virtue of the anticommutative relation in Eq.(62). In case when p = p', Eq.(63) becomes :
bpbp|0 = |2(p) = -bpbp|0 = -|2(p) ---------- (64)
which can be true only for |2(p) = 0. Therefore, two spin 1/2 particles cannot occupy the same state - a characteristics of Fermi-Dirac statistics.
In terms of the b, b operators, the probability density can now be re-defined as charge density. The total charge operator is in the form :
---------- (65)
where the dk's come from separating the Dirac spinors into 2 for the particle (electron) and 2 for the anti-particle (positron, according to Eq.(33)). The last term is the infinite negative charge of the Dirac sea to be subtracted manually.

Interaction between particles in quantum field theory is usually expressed by the interacting Hamiltonian. For example, the interaction between the pion (spin 0 particle) and nucleon (spin 1/2 particle) is HI = -g , where g is the coupling constant. Unlike non-relativistic quantum
Feynman Diagram theory in which the main concern is about energy levels, the quantum field theory works mostly with transition probability from an initial state i to the final state f, e.g., S = f | HI | i (known as the S matrix). Since the equation cannot be solved analytically with the addition of interaction, perturbation theory is employed to do the job approximately. Each term in the perturbation expansions can be represented by a

Figure 07e Nucleon Scattering [view large image]

 Feynman diagram, which has a mathematical correspondence to compute the S matrix. Figure 07e shows a simple Feynman diagram for nucleon-nucleon scattering via the exchange of virtual pion.

It may have been noticed that the anti-particle in the Feynman diagrams within Figure 03 is represented by an arrow pointing backward in time. This is because the signs of energy E and time t are inter-changeable without altering the uncertainty rule in Eq.(2). Therefore, an anti-particle is represented by an arrow backward in time instead of labeling it by -|E| every time. It doesn't mean that anti-particle can be used to do time travel back to the past.

Another weird phenomena in Figure 03 is the entanglement associated with the particle pair created in many of the processes. By the law of conservation of angular momentum, no matter how far apart are the members of this pair, if the spin is flipped for one of the member, the spin for the other member will also be flipped to the opposite at precisely the same moment. This non-local influence (non-locality) occur instantaneously, as if some form of communication which Einstein called a "spooky action at a distance", operates not just faster than the speed of light, but infinitely fast. Nonetheless, it is now a routine occurrence in quantum computing (see also the refutation of Einstein's objection by Bell's theorem).

See "The Different Perspectives of the 1st and 2nd Quantizations".

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