Quantization and Field Equation

Quantization and Field Equations

First Quantization and Schrodinger Equation
Klein-Gordon Equation of the Scalar Field
Derivation of the Dirac Equation
Interpretation of the Dirac Equation
Electromagnetic Wave Equation
Electromagnetic Wave Polarization and Photon Spin
Second Quantization and Feynman Diagram
A Footnote - Mathematics for Double-slit Experiment

First Quantization and Schrodinger Equation

Qunatum theory has its foundation in the uncertainty principle, which states in mathematical form :

p_x >

, where

denotes the uncertainty, x is the position of the point mass m along the x-axis, p_x = m v_x is the momentum along the x-axis, v_x is the velocity along the x-axis, and

= h/2

= 1.054x10^-27 erg-sec (Figure 01). A similar relation exists for the y and z components, and for the time t and energy E (including the rest mass energy mc²), e.g.,

E >

. The uncertainty

Figure 01 Uncertainty Principle

principle seems to be applicable up to a size of about 10^-7 cm - the size of the C₆₀ buckyball. In case of macroscopic objects with larger size and heavier mass, the uncertainties in position and time becomes very small, classical physics is applicable once more.

For example, the mass of the buckyball is about 10^-21gm, its velocity would be about 10⁴cm/sec at 400^oK, the uncertainty in its position should be greater than 10^-10cm, which is still many times smaller than its size of 10^-7cm. On the other hand a single carbon atom moving at such speed would have an uncertainty in position very close to its size of about 10^-8cm.

It can be shown that the expression for the uncertainty principle is equivalent to the commutative relations :
xp_x - p_xx = i

---------- (1),
Et - tE = i

---------- (2),
with a similar expressions for the y and z components. These formulas can be represented in terms of matrices, but it is more often to keep x, y, z, and t as ordinary c-numbers but the p_k's and E become non-commutative q-numbers in the forms of differential operators (the y and z components will be ignored now to simplify the formulation) :
p_x = -i

---------- (3a),
E = i

---------- (3b).
The partial differentiation acts only on the indicating variable while others variables are held constant. By substituting Eqs.(3a), (3b) into (1), (2) respectively and applying the equations to a function

(x,t), e.g.,
i

[- x + (x)]

= i

---------- (4)
it can be verified that the left hand side leads directly to the right hand side. The function

is called wave function. It is interpreted as the probability of finding the particle at (x,t) and thus a wave property to a particle. Dynamic of the wave function is to be determined from the energy equation for a free particle :
E = p²/2m ---------- (5)
which takes into account only the kinetic energy, the rest mass energy mc² is omitted in this equation. It is positively defined as the square of the momentum p is always positive. Substituting Eqs.(3a), (3b) into (5) we obtain the Schrodinger Equation for a free particle (ignoring the y and z components again):
-(

²/2m)

= i

---------- (6)
Solution for this partial differential equation can be obtained by the method of separation of variables, e.g.,

(x,t)=A(x)e^-iEt/, where the solution for t has been evaluated as an oscillating exponential function by virtue of the imaginary index, otherwise it will be an exponential function not in agreement with observation. Eq.(6) can then be reduced to an ordinary differential equation of the form :
(d²/dx²) = -(2mE/

²) ---------- (7)
It is similar to the equation of motion for the harmonic oscillator:
(d²/dt²)X = -KX ---------- (8)
with taking the role of X, x replacing t, and the force constant K = (2mE/

²). Thus the solution of Eq.(7) is another oscillating function :
= e^ikx ---------- (9)
where k = (2mE)^1/2/

. The final solution can be written as :

(x,t) = A e^-i(t-kx) ---------- (10)
where

=E/

. The integration constant A is determined by demanding that the probability of finding the particle within the extent of length L is 1, i.e.,
^*dx = A²L = 1 ---------- (11)
which yields A = 1/L^1/2. Thus the probability density ^* = 1/L, which indicates that the chance of finding the particle at any point within becomes smaller as the extension getting larger.
Eqs.(6) and (7) are linear differential equations because they involve only linear terms in the wave function. Linear equation has the special property that if ₁, ₂, ... are the solutions with k equals to k₁, k₂, ... respectively, the combined solution = A₁₁ + A₂₂ + ... is again a solution of the same differential equation, where the A_i's are interpreted in quantum theory as the probability amplitude of finding the particle in state k_i's and (A₁)*A₁+(A₂)*A₂+...=1. This is known as superposition, which forms the base in explaining interference and the production of wave packets, .... in wave theory. For example, superposition in the following form will generate a wave packet:

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

where f(k-k_o) is the weighting function specifying the k dependency of the coefficients A_k. Since the value of k runs continuously from -

to +

, the sum becomes integration over the k-space. In particular, when the weighting function is in the Gaussian form:

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

where

k is the spread of bell curve in k-space (it is related to the spread in x-space by the uncertainty relation

1), the wave packet can be evaluated explicitly :

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

Figure 02a shows pictorially the wave packet as a function of x (centering around x_o) and the weighting function (centering around k_o). A measurement on the wave packet will detect the particle at certain value of k. This is the Copenhagen Interpretation, which asserts that the wave function collapses to a single state in response to the measurement. In other word, the superposition is decoherent by the detecting apparatus such that the particle is not governed by the linear

Figure 02a Wave Packet
[view large image]

equation (e.g., by Eqs.(6) and (7)) anymore, and thus the break down (or collapse) of the superposition.

The wave-particle duality is usually demonstrated by the double slits experiment as shown in Figure 02b. A pattern consistent with the nature of particles emerges in the screen when the electrons go through only one slit. An interference pattern associated with

		the nature of wave appears as soon as the electrons are allowed to pass through both slits (even down to only one electron at a time). The experiment also shows that the properties of particle and wave cannot be displayed simultaneously. Mathematically, there are four terms in the double slits experiment - two "separate-slit" terms from slit 1 and 2, and two "interference" term between the waves from slit 1 and 2, i.e., P₁₂ = ₁^₁ + ₂^₂ + ₁^₂ + ₂^₁.
Figure 02b Two Slits Exp. [view large image]	Figure 02c Three Slits Exp. [view large image]	It is the latter two terms that produce the interference pattern, This is the Born's rule - a key law in quantum mechanics proposed in 1926. In plain language, it states that interference occurs in pairs of possibilities, higher order terms are ruled out. (see more in footnote)

In the 1990's, a certain theory of quantum gravity suggested that higher order terms such as

₁^*

₂

₃^*

₁ should be included in the three slits experiment (Figure 02c). Accordingly, an experiment was run in the late 2000's to check out the claim by subtracting the three slits patterns with the prediction from the Born's rule. The pattern is nullified within the 1% experimental error. Thus, the Born's rule has been vindicated for now. But the verdict is not final, there is room to improve the experimental uncertainty and the theoretical calculation should be reviewed for magnitude of the correction.

This version of quantum theory take into account only the mass of the particle, other physical attributes such as spin and charge do not appear in the formulation. It is also non-relativistic in the sense that the equation is not invariant under the Lorentz transformation. The underlying low energy context also implies that it is not capable of treating high energy phenomena such as pair creation and annihilation, production of new particles, ... Nevertheless by adding an interaction potential into the equation, this formulation has been applied successfully to the physics of molecules, atoms and nuclei. Another virtue is its definition of energy, which would never turns into an undesirable negative value (not observed in nature as particles are not falling into this negative energy hole). It is usually referred to as quantum mechanics in distinguishing from the fully relativistic quantum field theory.

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Klein-Gordon Equation of the Scalar Field

A relativistic theory still uses the same kind the canonical quantization rules as outlined in Eqs.(1) - (4). But the energy equation is now replaced by the one from the theory of special relativity, i.e.,
E² = (mc²)² + (pc)² ---------- (15)
In this formulation, (p_x, p_y, p_z, iE/c) together form a 4-momentum, which transforms like a 4-vector under the Lorentz transformation. The probability density has also to be redefined as the 4th (time) component of the flux density :

---------- (16)

which satisfies the continuity equation :

---------- (17)

where the Greek indices run from 1 to 4. By substituting Eqs.(3a) and (3b) into the energy equation Eq.(15) (ignoring the y, z components again) we obtain the Klein-Gordon Equation :

²[c² - ] = m²c⁴ ---------- (18).
The natural units are often adopted in particle physics with c =

= 1 by changing the unit of length to [1/(3x10¹⁰)]sec, and the unit of time to 1/(1.054x10^-27erg), then the unit of the 4-momentum, energy and mass becomes the inverse of length, while the charge, velocity, and angular momentum are dimensionless. The original value of c and

can always be re-introduced by dimensional consideration (see more in Natural Units). Thus, Eq.(18) can be re-written as :
[ - ] = m², ---------- (19)
the solution of which is :
= A e^-i(t-kx) ---------- (20)
where

=E, and k²=E²-m². Thus the 4-momentum can be written in the form : (k_x, k_y, k_z, iE). By re-arranging the energy equation into the form : E²=m²+k² and taking the square root :
E = (m²+k²)^1/2 ---------- (21).
It is immediately clear that we have a problem with the negative energy (as mentioned earlier). Another troublesome formula comes from the definition of probability density (via Eq.(16)) :
P = is₄ = 2AE ---------- (22)
which becomes negative for negative energy. But there is no such thing as negative probability density. The solution is to re-interpret P as the charge density, then particle with different sign of energy would associate with different charge leading to the particle/anti-particle duality. This logical step was not taken until 1934 by Pauli and Weisskoff after the discovery of positron in 1932. It was later shown in addition that the particle/anti-particle pair is represented by the and ^* fields respectively.

The Klein-Gordon equation was first considered as a quantum wave equation by Schrodinger in his search for an equation to describe the de Broglie waves. The problem with negative energy and the wrong prediction on the fine structure of the hydrogen atom leaded him to introduce the Schrodinger equation instead (omitting the fine structure). The equation was proposed by Klein and Gordon a year later in 1927 for a relativistic electron. It is now recognized that this scalar field equation describes a spin 0 particle such as the pion or the proposed Higgs particle. The fine structure of the hydrogen atom will be determined correctly by the Dirac equation for spin 1/2 particle.

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Derivation of the Dirac Equation

In 1928 P.A.M. Dirac tries to resolve the problem with negative energy by writing the equation in linear form (in natural units) :
m = bE - ap --------- (23)
where a and b are to be determined by demanding that the square of Eq.(23) takes the same form of Eq.(15), i.e., m²= E²-p². This scheme does not quite work since it is impossible to eliminate the cross terms between E and p. A more promising way is to break up the energy equation into the form such as Eq.(24) :
m² = [bE - (a_xp_x + a_yp_y + a_zp_z)][bE + (a_xp_x + a_yp_y + a_zp_z)] = b²E² - [(a_xp_x)² + (a_yp_y)² + (a_zp_z)²)] ---------- (24)
Beside the trivial case of a=b=1, a new formulation appears if the a_i's and b take the forms of the 2x2 Pauli matrices, and 2x2 identity matrix respectively (note that a_i²=1, and the Pauli matrices are more often denoted as

_i) :

---------- (25)

By substituting the canonical quantization rules of Eqs.(3a), (3b) into Eq.(24), we obtain (in natural units and ignoring the y, z components for the time being) :
[i( + a_x)][i( - a_x)] v = m² v ---------- (26)
where v is a two components single column matrix, e.g.,

v =

---------- (27)

Eq.(26) can be split into two :
[i( - a_x)] v = m u ---------- (28a)
[i( + a_x)] u = m v ---------- (28b)
where u is another two components column matrix similar to v. It is a mathematical property of the differential equations that more components will be created when transforms a second-order equation to two first-order equations, the components would be mixed between the first-order equations. Eqs.(28a) and (28b) are equivalent to the original equation written down by Dirac. The

Figure 02d Dirac Equation [view large image]

original form of the Dirac equation is obtained by adding and subtracting Eqs.(28a) and (28b) and then combining them into one first order differential equation in terms of the 4x4 gamma matrices operating on a 4-components single column matrix :

---------- (29)

where the gamma matrices

_i and the 4-components field (usually known as Dirac spinor)

can be expressed in the forms :

---------- (30)

and the repeated index in the expression indicates a sum. This is the Dirac equation cast in stone inside the Westminster Abbey (see Figure 02d). The problem with negative energy is still there, but the new form of equation turns out to be the one that governs all the particles in the material world, which consists of spin 1/2 particles, while particles of integral spin carry out interaction between the spin 1/2 particles.

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Interpretation of the Dirac Equation

To get around the problem of negative energy, Dirac proposed an energy spectrum containing all electrons in the universe (see Figure 03). In addition to the normal positive-energy spectrum, it also contains the negative-energy variety, which spans the spectrum from -m₀c² down to negative infinity. All the negative-energy levels are filled, thus the positive-energy particle is inhibited from transition into these lower energy states bypassing the unobserved phenomena of falling into the negative energy levels. Thus, the existence of negative energy has no observable effect in the real world. Only when there is enough energy available, e.g., when E

2m_oc², a real particle and anti-particle pair with positive-energy can be created from this unseen sea of negative-energy particles. The particle is the electron originally resided in the negative energy region, while the anti-particle (positron) can be interpreted as the hole in the vacated energy level acquiring a mass m_o. The law of charge conservation demands that this anti-particle carries a positive charge.

Figure 03 Negative Energy [view large image]

Many interesting things occur when the electron meets the hole (see Figure 03). Over the past three decades, the use of e⁺e^- collisions to probe the vacuum have yielded a great deal of information about the nature of the strong, weak, and electromagnetic interactions, and have played a major role in establishing the Standard Model of elementary particles (see more in "Quantum Vacuum"). Positron was first detected in 1932 by C. Anderson.

There are four types of solution for the Dirac equation depending on energy (positive or negative) and spin (up or down) as shown below :

---------- (31)

where V is the volume containing the free particle, the p and x are now 3 dimensional c-numbers, and the expressions have been reverted from the natural units. For the case of Dirac spinor, the probability density is the same as in the expression of Eq.(11) except that the wave function is now replaced by the 4-components Dirac spinor. In a procedure similar to the non-relativistic case, the normalization is obtained as shown in Eq.(31). Since the probability density is positively defined, it is not possible to have it re-interpreted as charge density, there is no change of sign by switching to negative energy (as in the case of the scalar field, see Eq.(22)). The correct formulation can be obtained only through the number operator via second quantization.

Charge conjugation is the operation that turns a field describing a certain particle into a field for the corresponding antiparticle. It can be shown that such operator is in the form :

---------- (32)

The antiparticle field is obtained by:

_C = eⁱC ^* ---------- (33)
which transforms ⁽¹⁾ to ⁽⁴⁾, and ⁽²⁾ to ⁽³⁾ (in Eq.(31)) as expected for the

_C's to be the negative energy solutions. The spinors here exclude the exponential e^i(px-Et), and

is a phase factor (0 and

respectively for the examples above).

According to the theory of electromagnetism a magnetic moment would be generated by any current loop. The Stern-Gerlach experiment demonstrated that the spinning electrons produce a pattern of two distinct parts corresponding to two opposite spin

orientations in the magnetic field. The fine structure of the hydrogen atom (1 energy level splits into 2) further determined that the spin of the electron is 1/2. In the presence of a magnetic field it is the z component which aligns with the field direction having a quantized value of

/2 while the entire magnetic moment precesses around the field with a magnitude S = [(s+1)s]^1/2 (Figure 04). The expression for the magnetic moment derived in quantum theory has the form:
M = (e/m)s ---------- (32)

Figure 04 Electron Spin [view large image]

where s = (

/2)(a_x + a_y + a_z), indicating that the a_i's are related to the spin of the electron.

In case the mass of the particle equals to zero, i.e, m=0, then Eqs.(28a) and (28b) are de-coupled. The 2-components spinors are often re-labelled as u = u_L and v = u_R, the equations can be re-written as :
ap u_L = -E u_L = -|p| u_L ---------- (35a)
ap u_R = E u_R = |p| u_R ---------- (35b)

Eq.(35a) describes a massless particle with direction of spin opposite to the direction of motion (left-handed helicity), while Eq.(35b) describes a massless particle with spin pointing to the same direction of motion (right-handed helicity). This kind of 2-components spinor is often referred to as Weyl spinor. They used to be associated with neutrino (only left-handed neutrino and right-handed anti-neutrino exist in nature) when it was considered as massless. The Weyl spinor has also been applied to the formulation of the Standard Model with u_L participated in both the electromagnetic and weak interactions, while u_R involves with only the electromagnetic interaction. Both the electron and neutrino are considered to be massless originally, only the electron acquires mass via interaction with the Higgs field in that model. The Weyl spinor also found its way into the theory of supersymmetry, which insists that every boson have a fermion partner (in the form of Weyl spinor).

Figure 05 Neutrino Mass
[view large image]

When neutrino has mass, its speed would always be lower than the speed of light, theoretically an observer can move in a speed faster than the left-handed neutrino, overtakes this neutrino and sees a right-handed anti-neutrino (Figure 05). That's why the two Weyl spinors mix in case of particle with mass as shown in Eqs.(28a) and (28b).

The 2-components spinor is related to the isotropic vector R=(r₁, r₂, r₃) with zero length, i.e.,
r₁² + r₂² + r₃² = 0 ---------- (36)
where the r_i's can be a complex number making a total of 6 dimensions. The R vector can be compressed into a 2-components spinor S=(s₀, s₁) by defining :
r₁ = s₀² - s₁² ---------- (37a)
r₂ = i(s₀² + s₁²) ---------- (37b)
r₃ = -2s₀s₁ ---------- (37c)
the inverse of which is :
s₀ = [(r₁ - ir₂)/2]^1/2 ---------- (38a)
s₁ = [(-r₁ - ir₂)/2]^1/2 ---------- (38b)

For example, let's take R=(1, i, 0), then S=(1, 0) or S=(-1, 0). If R is rotated by an angle

such that R => e^-iR around r₃, then S => e^-i/2S around s₁. That is, a rotation of R by an angle

translates to a rotation of S by an angle

/2. It is this peculiar prpperty of spinor that makes it so different from the other kinds of field. Particles associated with spinor are called fermion to distinguish themselves from the other types called boson (Figure 06).

Figure 06 Different Kinds of Spin [view large image]

This special property of rotation for spinor is also applicable to the internal (invisible) rotation in gauge transformation.

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Electromagnetic Wave Equation

The field equations for electromagnetic radiation is a product of the Victorian Era in the 19^th century. It was originated from a set of equations in electromagnetism. Maxwell's contribution is to add a "displacement current" term, i.e., the (1/c)E term in Eq.(40), and thus put the equations into a consistent set which implied new physical phenomena, at that time unknown but subsequently verified in all details by experiments. In terms of the electric field E and magnetic field B the Maxwell's equations are four first order differential equations :

where , the i, j, k, are unit vectors along the x, y, z axis respectively, J is the current density,

is the charge density, and c is the velocity of light. For regions of space where charge and current are absent, the right-hand sides vanish in all the equations. These first order differential equations can be translated into second order equations more suitable for describing the electromagnetic wave. By virtue of the mathematical identity

(

A)=0, Eq.(41) can be defined by the vector potential A:
B =

A ---------- (43)
By substituting Eq.(43) into Eq.(42) and using another mathematical identity ()=0, we can define E in terms of A and the scalar potential as :
E = -

- (1/c)A ---------- (44)
which combines with Eq.(39) yields:
² + (1/c)(

A) = 0 ---------- (45)
Substituting Eqs.(43), (44) into (40) and using one more identity

(

) A = -²A +

(

A), we get another second order differential equation :
²A - (1/c²)A -

(

A + (1/c)

) = 0 ---------- (46)
The definition of B in Eq.(43) is arbitrary to the extent that the gradient of of scalar function

can be added without changing anything by virtue of the identity () = 0. It means that the Maxwell's equations are unchanged by the transformation :
A

A' = A + ---------- (47)
which (via Eq.(44)) implies :

' = - (1/c)

---------- (48)

Eq.(45) is reduced to the Laplace's equation for time-independent electrostatic potential in free space, that's why it is also called "Coulomb gauge".
The longitudinal component is eliminated, i.e., there is no component along the direction of motion. This is ensured by the condition A' = 0.
The two transversal components correspond to the two polarizations as observed in electromagnetic radiation.
Since ' = 0 in free space without the presence of any source, the Maxwell's equations for electromagnetic wave become :
²A' - (1/c²)A' = 0 ---------- (49)
The magnetic field is related to A by B = A' ---------- (50)
The electric field is related to A by E = - (1/c)A' ---------- (51)
From Eqs.(49), (51), the wave equation for E can be derived :
²E - (1/c²)E = 0 ---------- (50a)
From Eqs.(49), (50), the transverse gauge, and the identity leading to Eq.(46), we obtain the wave equation for B :
²B - (1/c²)B = 0 ---------- (51a)
It is now clear that the introduction of the vector potential A is mainly to simplify computations. For the case of transverse electromagnetic wave two components of A is required to describe the wave, while the E and B fields together introduce four components.

Eqs.(49), (50a), (51a) are in a form very similar to the Klein-Gordon equation in Eq.(18) except that the mass term vanishes (because the photon has no rest mass) and the field is a vector (instead of scalar) with two transversal components (polarization) perpendicular to each other. The Maxwell's equations were derived more than one hundred years ago without any knowledge about quantum theory, the similarity in form is perhaps because all kinds of wave (including the gravitational wave for spin 2 particle) are governed by the same kind of equation. Only the details (such as number of components) and interpretations are slightly different (note the omission of the Planck constant h in the Maxwell's equations has no consequence in the comparison as m = 0).

The solution of Eq.(49) is :

k_x, k_y, k_z = 2n

/L (n = 1, 2, ...) with k

x = k

x -

t = k

x - |k|ct. The c_k coefficients is linked to the spectrum associated with the electromagnetic radiation. It will assume the role of q-number in second quantization, which endows particle attributes to the wave. While the vector potential A has dimension (erg/cm)^1/2, the dimension for the c_k is erg^1/2-cm.

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Electromagnetic Wave Polarization and Photon Spin

		For coherent EM wave propagating along the +z axis (Figure 07a), the two transversal electric fields can be expressed as : E_x = E_x0 cos(t - kz + _x) ---------- (53a) E_y = E_y0 cos(t - kz + _y) ---------- (53b) where E_x0, E_y0 are amplitudes, and _x, _y are the phase angles of the x, y components respectively (Figure 07b). These equations can be simplified somewhat if the view is facing the x-y plane fixed at z=0 (such as the view in Figure 07c), and by using the relative phase angle = _x - _y :
Figure 07a Polari-zation	Figure 07b Phase Angles [view large image]	E_x = E_x0 cos(t) ---------- (53c) E_y = E_y0 cos(t - ) ---------- (53d)

By defining an angle in term of the ratio of the two components (Figure 07a) :

= tan^-1(E_y/E_x) = tan^-1{(E_y0/E_x0)[cos(

) + tan(

t)sin(

)]} ---------- (53e)
the type of polarization can be readily discerned by the values of E_x0, E_y0, and

as shown in Table 01 below.

Phase Angle	tan()		Polarization
0^o	0	0^o	Linear along x-axis
0^o		90^o	Linear along y-axis
0^o	1	45^o	Linear at 45^o
45^o	(1/2)^1/2(E_y0/E_x0)[1 + tan(t)]	Right-handed rotation	Elliptical or circular
90^o	(E_y0/E_x0)[tan(t)]	Right-handed rotation	Elliptical
90^o	tan(t)	t	Right-handed Circular for E_y0=E_x0
180^o	0	0^o	Linear along x-axis
180^o	-	-90^o	Linear along y-axis
180^o	-1	-45^o	Linear at -45^o
-90^o	-(E_y0/E_x0)[tan(t)]	Left-handed rotation	Elliptical
-90^o	-tan(t)	-t	Left-handed Circular for E_y0=E_x0

Table 01 Types of Polarization (from formula)

Table 01 shows that for linear polarization the angle

is constant with E_x and E_y varying in such a way that its ratio is still a constant. For circular polarization the length of the field |E| is constant, but

changes with a frequency determined by

. For elliptical polarization even |E| is not a constant. Another way to check out the polarization is to view the animation provided by Amanogawa in Figure 07c. Table 02 shows the types of polarization associated with different sets of parameters in the application.

Figure 07c Polarization of EM Wave [see animation]

Amplitude x	Phase x	Amplitude y	Phase y	Polarization
1.0	0^o	1.0	0^o	Linear at 45^o
1.0	0^o	0.0	0^o	Linear along x-axis
0.0	0^o	1.0	0^o	Linear along y-axis
1.0	45^o	0.0	0^o	Linear along x-axis
1.0	45^o	1.0	0^o	Right-handed elliptical
0.5	0^o	1.0	0^o	Linear at 63.4^o
0.5	30^o	1.0	30^o	Linear at 63.4^o
1.0	0^o	1.0	90^o	Left-handed circular
0.2	0^o	1.0	90^o	Left-handed elliptical

Table 02 Types of Polarization (from animation)

Linear polarization is characterized by the relative phase angle

= 0, the direction of oscillation

depends on the ratio E_y0/E_x0. Let's take E_y0=E_x0 to simplify the formulas, then the polarization vector can be expressed as :
|e> = cos

|e_x> + sin

|e_y> ---------- (53f)
where |e_x> = and |e_y> = are the unit vector in the x (upper component) and y (lower component) directions respectively. It is similar to the Jones vector in classical electrodynamics and the polarization vectors in Eq.(52). Since the unit vectors satisfies the orthogonality relation : < e_i|e_j> =

(i-j) for (i, j) = (x or y), < e|e> = 1, if it is interpreted as the total probability, then cos²

and sin²

can be interpreted as the probability in polarization state |e_x> or |e_y> respectively.

For

= 90^o or -90^o,

= +

t (right-handed circular polarization) or -

t (left-handed circular polarization). The polarization vector in Eq.(53f) can be expressed in the form :
|e> = (e^-it/) |e_L> + (e^it/) |e_R> ---------- (53g)
where
|e_L> = (|e_x> - i|e_y>)/ = / ---------- (53h)
|e_R> = (|e_x> + i|e_y>)/ = / ---------- (53i)
are the unit vectors for circular polarization to the left and right respectively. Thus, circular polarization can be expressed in terms of the combination of linear polarizations and vice versa as shown below :
|e_x> = (|e_L> + |e_R>)/ ---------- (53j)
|e_y> = i(|e_L> - |e_R>)/ ---------- (53k)
Since the circular unit vectors again satisfies the orthogonality relation : < e_i|e_j> =

(i-j) for (i, j) = (L or R), < e|e> = 1. If it is interpreted as the total probability, then there are 50% probability for each of the circular polarization state in Eq.(53g). In order to obtain a definite circular polarization, we have to pick either +

t or -

t but not both. A linear polarization will be produced if we keep both +

t and -

t in Eq.(53g). For example, the purely right-handed rotating polarization vector is described by :
|e> = e^it |e_R> ---------- (53l)

The angular momentum density of classical electromagnetic waves is :
S = r X [E(r,t) X B(r,t)]/(4

c) ---------- (53m)
where r is a positional vector pointing out from the origin of the coordinate system. It can be shown that a circularly polarized plane wave moving in the z direction has a finite extent of the field in the x and y directions (Problem 6.11, Classical Electrodynamics, by J. D. Jackson, 1967). Such spatial extension has a direction associated with either |e_R> or |e_L>. Averaging over space and time by one wavelength Eq.(53m) yields :
S = U/|

| ---------- (53n)
where U=|E|²/(8

) is energy of the wave per unit volume, the + sign is for right-handed circular polarization, the - sign for the left-handed one. An experiment back in 1936 had demonstrated conclusively that circular electromagnetic wave does carry angular momentum.

Transition to quantum theory is accomplished by expressing U=N

/V, where N is the number of photons in volume V, and the energy has been quantized to

. Thus, the angular momentum of a photon can be parallel or anti-parallel to the z-axis :
s_z =

---------- (53o)

This formula shows that angular momentum of electromagnetic wave is related to the right- or left-handed polarization, which becomes photon spin of +

or -

respectively when goes over to quantum theory, i.e., the photon is spin 1 particle. It is said to have right-handed helicity if the spin aligns with the direction of motion (+

), or left-handed helicity in opposite direction (-

) as shown in Figure 07d. As usual in quantum theory, the photon would be in a superposition of both the right- and left-handed helicity states, it is only when a measurement is performed that it assumes a definite right- or left-handed helicity state (see Copenhagen Interpretation).

Figure 07d Photon Helicity [view large image]

For linear polarization, although the average of large number of photons shows zero angular momentum, individual photon does carry either right-handed or left-handed polarization with 50% probability for each as shown in Eq.(53g).

[Top]

Second Quantization and Feynman Diagram

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 of Eq.(20) :

----------(54)

where k is the 4-momentum (k_x, k_y, k_z, 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)

If a number operator N_k = c_k^*c_k is defined such that it operates on the state vector |n_k

to generate:
N_k|n_k

= n_k|n_k

where n_k is the number of particles in the k state; it follows that
c_k^*|n_k

= (n_k+1)^1/2|n_k+1

c_k|n_k

= n_k^1/2|n_k-1

Thus c_k^* increases the number of particles in the k state by 1, while c_k 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 k^l and n_l. 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 c_k^*c_k'^*|0

= |1(k),1(k')

= c_k'^*c_k^*|0

= |1(k'),1(k)

---------- (59)
by virtue of the commutative relation in Eq.(56). In case when k = k', Eq.(59) becomes :
c_k^*c_k^*|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 (A)_ij = (A*)_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 :
b_p^*b_p'^*|0

= |1(p),1(p')

= -b_p'^*b_p^*|0

= -|1(p'),1(p)

---------- (63)
by virtue of the anticommutative relation in Eq.(62). In case when p = p', Eq.(63) becomes :
b_p^*b_p^*|0

= |2(p)

= -b_p^*b_p^*|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 d_k'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 H_I = -g , where g is the coupling constant.

	Unlike non-relativistic quantum 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 \| H_I \| 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
Figure 08 Nucleon Scattering [view large image]	approximately. Each term in the perturbation expansions can be represented by a Feynman diagram, which has a mathematical correspondence to compute the S matrix. Figure 08 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).

[Top]

A Footnote - Mathematics for Double-slit Experiment

Here's more details to expand the hand-waving statement in the main text. All the assumptions usually omitted in textbooks will also be clarified in the process. We begin with the Huyghens' principle, which is applicable to electromagnetic wave as well as matter wave (such as electrons). It states that if we know the value of the wave function on a given wave front, then we can express its value elsewhere as the sum of contributions from different elements of the wave front in the form of propagating spherical wave,

	$Diffraction Pattern$	e^{(2i r/)}/r, where r is the distance from the point in question to the element of surface on the wave front, and is the wavelength. This spread of wave is somewhat similar to the pattern (in the wall behind) generated by firing a machine gun at an iron plate with two slits on it. Most of the bullets through the slits would align with each, some may be scattered sideways by hitting the edges although the chance becomes rarer as deviation getting further. One big difference is that bullets don't interfere with each other (story ascribed to Feynman). Anyway, assuming that the slit is very narrow, the two waves from slit 1 and 2 can be represented as: ₁ = Ae^{(2i r₁/)}/r₁ ---------- (66a) ₂ = Ae^{(2i r₂/)}/r₂ ---------- (66b)
Figure 09 Double-slit Exp. [view large image]	Figure 10 Probabilities [view large image]	where A is the normalization constant, r₁, and r₂ are the distance from the slits to the screen as shown in Figure 09, which lays out the x-y plane only, while the z direction is perpendicular to the

computer screen. The expression for finite slit size is more complicated but will produce essentially the same result. Since the spacing between the slits is small comparing to the size of the apparatus, i.e., D >> d, therefore, r₁

r₂ and they are lumped together as r (see Figure 09). However the tiny difference d sin

(in the interference term) is crucial in producing the pattern (see Figure 09 again). Thus, the probabilities can be expressed as :

₁^*

₁

₂^*

₂

(A²/r²) ---------- (67a)

₁^*

₂ +

₂^*

₁

(2A²/r²) cos[2

(dsin

)] ---------- (67b)
The total probability is : P₁₂(

) =

₁^*

₁ +

₂^*

₂ +

₁^*

₂ +

₂^*

₁ = (4A²/r²) cos²[

(dsin

)] ---------- (68)
The normalization constant A can be evaluated by integrating Eq.(68) numerically over the angle

from -

/2 to +

/2 :
1 = (8A²/D²)cos²

cos²[

(dsin

)] d

---------- (68a)
For the values of d and

quoted below, the integral is equal to about 0.390, which gives A = 0.566D. For rapid oscillation of the interference term, i.e., when d >>

, the integral approaches a value of

/8, giving A² = D²/

or A = D/

^1/2 = 0.564D. Then it shows clearly that P₁₂(

) can be interpreted as the probability per unit radian.

It is impossible to produce a graph with realistic data for light wave using home computer and Basic programming. Figure 10 is produced for some sort of matter wave with

= 0.001 cm, d = 0.005 cm, and D = 10 cm. The "separate terms" together is plotted in green, the "interference terms" in blue, and the total probability is in red. The only running variable in the numerical computation is sin

, which varies from -0.8 to +0.8 in steps of 0.001. Since cos²

= 1 - sin²

, tan

= sin

/cos

, from which we can compute r = D/cos

, x = D tan

. It is rather obvious from Eq.(68) that maxima occur at :
d sin

= n

, where n = 0, 1, 2, 3, ...
and minima occur at :
d sin

= (n/2)

, where n = 1, 2, 3, ...

The avid readers may have already noticed that the probability goes negative in the interference terms every half oscillating cycle (see Eq.(67b) or the blue curve in Figure 10). One explanation assures us that only the total probability P₁₂ is observed, we should not worry about how the various terms added up as long as P₁₂ can never be negative as shown in Eq.(68). It's somewhat like borrowing money from the bank, the account may become negative for awhile, but that's alright as long as payment is made on time. Or as explained by Feynman : no one objects to use negative numbers in calculations, although "minus three apples" is not a valid concept in real life.