The electromagnetic field is represented by Aμ which is related to the classical concepts of the electric and magnetic fields in the following way. You define the following antisymmetric tensor.
Fuv = [partial derivative]u Av - [partial derivative]vAu
where
F0i = [partial derivative]0 Ai - [partial derivative]iA0 = -Ei
Fij = = [partial derivative]i Aj - [partial derivative]jAi = [epsilon]ijkBk
where E and B are the electric and magnetic fields. The Lagrangian for electromagnetism is
L = -(1/4) Fuv Fuv - JuAu
Electromagnetism has gauge freedom. It is not affected by adding a scalar field.
Au -> Au + [partial derivative]u [psi]
where ψ is any scalar field, and has no effect. It does not alter the field tensor.
Fuv = [partial derivative]u Av - [partial derivative]vAu
Therefore, it is always possible to make a scalar field vanish.
For a massive particle, E = mc2. In naturalized units, where c = 1, E = m. Therefore, mass and energy are interchangeable, and you are free to measure the mass of particles in units of energy. For photons, you are supposed to use the wave form, where E = [h bar]w, where w is the angular momentum. Also, p = [h bar]k where p is the momentum and k is the wave number. The wavenumber k = 2π/λ where λ is the wavelength. In naturalized units where [h bar] = 1, then E = w, and p = k, so for photons, energy and angular momentum are interchangeable, and momentum and wave number are interchangeable. In quantum field theory, you often see the following relation.
ei(kx-wt)
This is called the plane wave, and is the wavefunction of a particle traveling in a straight line. If you do a Fourier expansion of an electromagnetic wave, you get two such things, with coefficients that are related to the annihilation and creation operators. These create and destroy particles, in this case photons, which relates the wave form to the particle form. This was first done by Max Born. You end up quantizing the field, which is why it's called quantum field theory or second quantization.
Let's say you have electromagnetic radiation in a box of volume V, where L is the length of a side, and V = L3. Now do a Fourier expansion.
A = 1/[squareroot of V] [summation of k, [alpha]] [c[epsilon](alpha)ei(kx-wt) + c*[epsilon](alpha)e-i(kx-wt)]
where the expansion coefficients c and c* are functions of the wavenumber k and α. α is an index labeling the polarization, which is described by two unit vectors, ε1 and ε2, which are perpendicular to k and to each other. The Hamiltonian of the system is
H = ½ [integral] (B2 + E2)dV = ½[integral][([nabla] x A)2 + [A dot]2]dV
You find that each half of the Hamiltonian gives the same result so that the total is
H = [summation of k, [alpha]] w2 (cc* + c*c)
You can also take into account the possibility that the c's depend on time by the redefinition
c(t) = ce-iwt
Now let's define the new variables.
q = c + c*
p = (w/i)(c - c*)
Then the Hamiltonian is
H = [summation](1/2) (p2 + w2q2)
This is just the Hamiltonian of harmonic oscillations, which is what you would expect since the electric and magnetic fields are harmonic oscillators.
Now p and q are operators acting on a wavefunction, and obey the following commutation relation.
[q, p] = i[h bar]
Now let's define the following ladder operator.
a = [squareroot of (2w/[h bar])]c
Then you end up with
H = [summation]([h bar]w/2)(aa† + a†a)
H = [summation](N + (1/2))[h bar]w
where [a, a†] = 1, and the number operator N = a†a.
The number operator N = a†a is Hermitian, and you have
N | n > = n | n >
You have the following commutators.
[a, N] = a
[a†, N] = -a†
a is the annihilation operator, and a† is the creation operator.
a | n > = [squareroot of n] | n - 1 >
a† | n > = [squareroot of n + 1] | n + 1 >
Now look at the following equation.
H = [summation](N + (1/2))[h bar]w
N is the number of particles, and see that even when N = 0, H is not zero. This is the origin of the idea that the vacuum itself has energy. The energy density of the vacuum is given by
pvac = ([h bar]/2[pi]2c5) [integral] w2dw
which, according to this, would be infinite. You can make it finite by assuming that this is a finite instead of infinite integral.
Now, you can think of the vacuum energy as caused by virtual photons, and if you have smaller and smaller waves of virtual photons going down to infinity, then that would be an infinite amount of energy. If there was some lattice that prevented such arbitrarily small waves, then the energy would be finite. That's what you have inside a crystal in solid-state physics, and in that case, the equations predict what you observe. Now space itself could be quantized and form such a lattice but it would have to be at a smaller scale than we observe, since we don't observe any such thing, and if that were true, the value of the vacuum energy, or zero point energy, would be 10120 times higher than we currently observe. We currently have no explanation for this discrepancy. We can observe a small vacuum energy called the Casimir effect. If you have two parallel plates, only an integral number of standing waves of virtual photons can exist between the plates, while those on the outside have no such restriction. This causes a tiny force pushing the plates together, which has been experimentally detected. The Casimir effect was predicted by the Dutch physicist Hendrick Casimir in 1948, and was experimentally measured by Steven Lamoreaux in 1996.
I assume the reader has already read my other papers on tensors and Lagrangians. Even if you have already read them, you might want to look over my paper on Lagrangians to refresh your memory. The Lagrangian L is the kinetic minus potential energy. The integral of Ldt is the action S, and you choose the function x such that S will be stationary for small changes in x. For a more detailed account, read my paper on Lagrangians. This is the Lagrangian for an electromagnetic field.
L = -(1/4) Fuv Fuv - [mu]0JuAu
where
Fuv = [partial derivative]u Av - [partial derivative]v Au
This is the Lagrangian for a fermionic field
L = i[psi bar] [gamma]u [partial derivative]u [psi] - m[psi bar] [psi]
Now the really neat thing is all you have to do is add these two Lagrangians together to get the interaction between an electromagnetic and fermionic field.
L = Lf + Lem = = i[psi bar] [gamma]u [partial derivative]u [psi] - m[psi bar] [psi] - (1/4) Fuv Fuv - JuAu
Here the 4-current Ju is the quantum 4-current from the fermionic field, which is charge times the probability 4-current.
Ju = e [psi bar] [gamma]u [psi]
This is the interaction term. It describes the mixing of the two fields. If you think of fields as sums of creation and annihilation operators, then these mixed terms allow you to change the number of two species simultaneously. This resulting theory is called quantum electrodynamics or QED.
To use the QED Lagrangian to calculate things, you need the perturbation to the Hamiltonian. If you make a perturbation to the Lagrangian, L → L + ΔL, since
H = p[q dot] - L
the perturbation to the Hamiltonian is ΔH = -ΔL so
[delta]H = e [psi bar] [gamma]u [psi] Au
If I were to approach quantum field theory rigorously, I would first discuss φ4 theory, which is the simplest quantum field theory, having only scalar fields. The two-point correlation function or two-point Green's function is <Ω|Tφ(x)φ(y)|Ω>, where |Ω> is the ground state of the interacting theory, which is different from |0> which is the ground state of the free theory. T is the time-ordering symbol. The correlation function can be interpreted as the amplitude for the propagation of a particle or excitation from y to x. If you have <0|Tφ{(x1)φ(x2)...φ(xn)}|0>, which is the vacuum expectation value of time-ordered products of a finite number of free field operators, you can greatly simplify the calculations, which involves putting them in all possible combinations, by using Wick's Theorem. From this, you can determine all possible Feynman diagrams of an interaction.
In particle physics, you often describe scattering events, where particles enter from infinity, interact, and then fly off. The transition between different states can be described by the following operator.
[psi] (t) = U(t, t0) [psi] (t0)
where the transition operator is the S-matrix.
U([infinity], -[infinity]) = S
The U operator satisfies the Schrodinger equation which is integrated to yield the following equation.
U = 1 - (i/[h bar]) [integral to t0 to t] HI Udt
where HI is the interaction term of the Hamiltonian. H = H0 + HI The U matrix is unitary. U-1 = U†
As you see, U appears in its own definition, so you can't solve for S exactly but it can only be solved iteratively by continually plugging in the results to get a better approximation.
S = 1 - i[integral]HIdt + (-i)2[integral]HI(t1)dt + [integral from -infinity to t1]HI(t2)dt + …
A version of the S-matrix called the bootstrap method at one time was an alternative to the theory of quarks, and gave rise to Regge theory. The theory of quarks and the Standard Model won that argument. However, Regge theory helped inspire early string theory, which is now our way of viewing the Universe.
The interaction Lagrangian is a mixture of creation and annihilation operators. For quantum electrodynamics, you have
LI = e [psi bar] [gamma]u [psi] Au
The successive higher orders of perturbation theory represent the creation and annihilation of more and more particles as part of the intermediate stage of the scattering process.
An easy way to write down a given scattering process, and the creation and annihilation of particles, is by Feynman diagrams. The interaction Hamiltonian for QED describes the creation and annihilation of electrons, positrons, and photons. There are eight possible combinations which are drawn below. Here I've drawn t as the horizontal axis, and x as the vertical axis, although sometimes it's shown reversed. Arrows pointing towards increasing time are electrons, arrows pointing backwards are positrons, and wavy lines are photons.
Now, your gut feeling is to imagine these as pictures of physical particles flying through space, interacting, and flying off. That's not really what it is. Remember that in quantum mechanics, what really exists when you're not measuring it is the wavefunction, which can be identified with the probability of detecting a particle. Here, the creation and annihilation operators describe the excitation of plane wave states rather than particles. The spacetime coordinates are variables to be integrated over rather than the positions of particles. Really, these diagrams are a non-algebraic representation of the S-matrix element.
None of these eight diagrams are possible all by themselves since they violate conservation of energy-momentum. You need two diagrams together so the failure of energy-momentum conservation in one diagram is cancelled out by that of the second.
Compton scattering is e- γ → e- γ. Moller scattering is e- e- → e- e-. Bhabha scattering is e+ e- → e+ e-. Rutherford scattering is the scattering of an electron by a proton or neutron. Electron-positron annihilation is e+ e- → 2γ.
Here are some examples of such processes. Here is Compton scattering, where there is an electron and photon in both the initial and final states.
Here is the diagram for Moller scattering where there is two electrons in both the initial and final states.
A Feynman diagram has external lines, which are particles that either enter from or exit to infinity, and internal lines, which are particles that exist only inside of the reaction. These internal particles are called virtual particles. You see that in Moller scattering, the two electrons exchange a virtual photon. Electromagnetism is nothing more than the exchange of virtual photons between charged particles. The other forces are also nothing more than the exchange of virtual bosons between particles. The strong force acts by the exchange of virtual gluons. The weak force acts by the exchange of virtual W+, W-, and Z0 bosons. Gravity acts by the exchange of virtual gravitons.
Virtual particles do not obey the same rules as normal particles. The energy and momentum of normal particles obey the mass shell condition.
Pu Pu = m2
This is not true for virtual particles. Unlike a normal photon, a virtual photon will not satisfy E = pc.
However, many popular accounts of virtual particles are inaccurate. It is often said that virtual particles are created by borrowing energy. Actually, energy is conserved at the vertex. Also, in many books, it says that virtual particles can only be exchanged over subatomic distances. Actually, virtual photons are massless, and thus of infinite range. This should hardly be surprising since otherwise electromagnetism would not be a long range force. If a compass needle points north, that means that virtual photons are traveling tens of thousands of miles from the tip of the compass needle to the magnetic north pole of the Earth. If Earth orbits the Sun, instead of flying off at a tangent, that means it's gravitationally bound to the Sun, which means that virtual gravitons are traveling 93 million miles between the Earth and the Sun.
One explanation of this is based on the Uncertainty Principle derived by Werner Heisenberg in 1927, where
[delta]x [delta] p > [h bar]
where Δx is the uncertainty in position, and Δp is the uncertainty in momentum. According to this, creation of a massless particle such as a photon requires a tiny amount of energy and momentum. Thus Δp is very small, and Δx is very large, which causes it to be a long range force.
The beauty of Feynman diagrams is that it allowed you to quickly write down any interaction algebraically, and solve scattering problems. Before Feynman diagrams, only an elite group of specialists worked with these sort of problems. Feynman diagrams opened up QED to the masses, comparatively speaking, in that any physicist could now do these problems. You define a covariant scattering matrix Mf that factors out the general features of the problem from the S-matrix.
Sfi = [delta]fi + (2[pi])4 [delta](4) (Puin - Puout) [squareroot of F(E)] Mfi
where Pu is the 4-momentum summed over all particles, and the function F(E) accounts for all normalization constants in the field expansion.
F(e) = [pi]in (2VE)-1 [pi]out (2VE)-1 [pi]f (2w)
where the last term is for fermions.
Feynman realized you could write down the invariant amplitude M just by looking at the diagram. The various factors are multiplied together, and integrated over the 4-momenta of the internal lines.
M = (2[pi])-4 [integral] _____d4q
In the blank, you multiply together the terms from the Feynman rules.
1. For each vertex, write ie[gamma][alpha]
2. For each internal photon line of 4-momentum k, write (-ig[alpha][beta])/k2 + i[epsilon])
3. For each internal fermion line of 4-momentum p, write i/([p slash]- m + i[epsilon])
4. For each external input electron, write ur(p)
5. For each external output electron, write [u bar]r(p)
6. For each external input positron, write [v bar]r(p)
7. For each external output positron, write vr(p)
8. For each external input or output photon write [epsilon]r[alpha](k)
9. For each closed fermion loop, multiply by -1, and take the trace over spinor indices, meaning summing over spin states of the virtual electron-positron pair.
Up till now, we have only considered the first terms in the S-matrix which give rise to tree level diagrams without loops. Here is the first order approximation for Moller scattering, where two electrons exchange a photon.
However, this is not the only possible diagram that could explain this reaction. Here are two second order diagrams.
Indeed, there are an infinite number of possible diagrams. In order to truly calculate cross sections or anything else about these or any other particle interactions you would have to sum the results from an infinite number of Feynman diagrams, which is clearly impossible. However, this is not as serious a problem as it would first appear. Notice that the first order diagram has two electron-electron-photon vertices, while the second order ones have four each. For each such vertex, you multiply by e. In general, for n photons, you multiply by e2n. Therefore, each term in the perturbation series is about e2 = 1/137 times smaller than the one before it. Therefore, you can just look at the first order approximation, or at most second order, and have a very good approximation.
In 1947, Lamb and Rutherford found that the s state of the hydrogen atom had slightly higher energy than what was predicted by quantum mechanics. This is called the Lamb shift, and is explained if you include the one loop corrections in the Feynman diagrams.
However, there is a deeper problem with QED than this. Let's look at this diagram.
Due to the conservation of energy-momentum, the momentum of the photon is divided between the electron and positron on the internal loop. There are an infinite number of ways it can be divided between them, and you have to take them all into account. You have to sum over the infinite range of momenta of the two particles. You can perform integrals over the infinite range of loop momenta, but when you do that, the integrals diverge.
The solution to this problem is called renormalization, and was developed in the late 1940's by Sin-Itiro Tomonga, Julian Schwinger, and Richard Feynman, who shared the 1965 Nobel Prize for their work. An example of renormalization is that an electron is surrounded by a cloud of electron-positron pairs that partially cancel its electric charge. Thus as you get closer to an electron, you partially penetrate this cloud, and its measured value of its electric charge increases. You could say its bare charge is infinite, and there is an infinite number of electron-positron pairs, so their effect is also infinite. These two infinities cancel each other out, leaving the measured value of an electron's charge from a distance. Essentially two values which you have no way of knowing, and are in some sense mathematically infinite, combine to give a value you can measure. In this way, infinities are absorbed into physical constants, and QED is made to work. Another example is the electron's measured mass which is the bare mass plus the self-energy which is the result of the electron continually admitting and reabsorbing photons.
The problems associated with renormalization are related to the vacuum energy. A theory that produces infinities when no particles at all are present, can hardly be expected to behave better in a more complicated situation. Despite these difficulties, renormalization is more easily dealt with in QED than anywhere else in particle physics.
You can think of renormalization the following way. You only calculate the theory to a certain energy or distance called the cut-off. As you let the cut-off distance go to zero, the predicted bare values of various quantities change and might go to infinity, which is fine since it's assumed that quantum effects which we don't know about will combine with the bare values to create the measured values we know about. You might have quantities whose measured value is zero change to non-zero. You just have to make sure you have enough parameters to begin with. If you only need a finite number of parameters, it works out, and the theory is renormalizable. If you need to add a infinite number of parameters, it does not work, and the theory is nonrenormalizable. Another thing you can do is the renormalization group or regularization. Here you are changing not the cut-off distance but the distance at which you are making the measurements, and letting that go to infinity. Often, nonrenormalizable terms in the theory will go to zero at long distances. Therefore theories that we use all the time, and work fine, might break down at very high energies or small distances, but we don't have to worry about that, because they work fine at the energy scales at which we use the theory. In some interpretations of string theory, there are not four but an infinite number of forces, but all but four are negligible at less than the highest energies, and so we don't know about them.
Richard Feynman also came up with a path integral formalism where the integral kernel, or propagator, of the time evolution operator can be expressed as a sum over all possible paths between two points.
The Standard Model rests on the assumption of gauge symmetry. This is where if you do a transformation, it doesn't affect the validity of the results. A gauge symmetry indicates a hidden degree of freedom. In electromagnetism, you can make the following transformation.
A -> A + [nabla]X
In quantum mechanics, the Lagrangians describing a quantum field will be invariant under the transformation.
[psi] -> [psi]ei[theta]
where θ is any constant angle. This is called global gauge symmetry. However, in quantum theory, the phase should be locally unobservable, meaning hidden. Therefore, it should be invariant under the following local gauge transformation.
[psi] -> [psi]ei[theta]xu
However, the Lagrangians do not initially appear to be invariant under this transformation.
Let's say you have the Dirac Lagrangian.
L = i[psi bar][gamma]u [partial derivative]u [psi] - m2 [psi bar] [psi]
Under the local transformation, it will transform as
L -> L - [psi bar] [gamma]u ([partial derivative]u [theta]) [psi]
The gauge transformation will have observable consequences. The problem is you have terms with the derivative of θ. You want to get rid of them the same way you could if θ was a constant. The way around this is to add a term whose purpose is to cancel out the unwanted θ terms. You replace ∂u with ∂u + Vu
[partial derivative]u with [partial derivative]u + Vu
Then you make the transformations.
[psi] -> ei[theta] [psi]
Vu -> Vu - i[partial derivative]u [theta]
Then the terms cancel, and you have local gauge invariance. However, in order to do this, you have to add a new field Vu. You don't want this new field to spoil the gauge invariance, so its derivatives must occur in the antisymmetric combination.
Fuv = [partial derivative]uVv - [partial derivative]vVu
so the simplest kinetic energy term is FuvFuv.
Thus the resulting Lagrangian for a fermion field is
L = I[psi bar] [gamma]u Du [psi] - m2 [psi bar] [psi] - (1/4)FuvFuv
where
Du = [partial derivative]u + ieAu
The new field is essentially the QED Lagrangian. Notice how this resulted automatically from making the Dirac equation locally invariant. The field we invented just to cancel unwanted terms turned out to be electromagnetism. This is what you have throughout the Standard Model. With the other forces, you introduce a vector field to get rid of the 4-vector ∂u. This creates new terms in the Lagrangian that are to be identified with the particle that mediates the interaction. Because they are vector fields, the particles are spin-1 gauge bosons. They arise automatically as a result of making the Lagrangian locally invariant.
In the 1920's, physicists had a semi-classical view of the atom, based on analogy with the Solar System. In the Solar System, the planets orbit the Sun, and also rotate on their axes. Similarly, in the Bohr model, the electrons were imagined to orbit the nucleus, and also spin on an internal axis like a top. However, since this was semi-classical, only certain values were allowed. Angular momentum about the nucleus was quantized in integer multiples of [h bar], and angular momentum about their axis, or spin, was quantized in half-integer multiples of [h bar]. Angular momentum, L, and spin, S, were two of the quantum numbers that identified an electron in an atom, the third being what electron shell it's in. Setting [h bar] = 1, the spin of an electron could be either + ½, 0, or - ½. The intrinsic spin of a particle is the absolute value of its maximum, so an electron has a spin of ½. Particles with integer spin are called bosons. Particles with half-integer spin are called fermions. Fermions obey the Pauli Exclusion Principle, while bosons do not. The reason is because fermions have antisymmetric wavefunctions which cancel out when there's more than one, while bosons have symmetric wavefunctions.
An electron has to rotate 720° to return to the same state. Most people assume this has something to do with the weirdness of quantum mechanics but actually that's not true. There is an entirely classical type of spin where something has to rotate 720° in order to return to its original state, and this is described mathematically by a spinor. This can be demonstrated by several parlor tricks, such as Dirac's scissors, Dirac's belt, Dirac's candle, Dirac's teacup, etc. The spin of a particle is also described by the same mathematical entity.
It should be emphasized that the Solar System model was abandoned in the 1930's. Names such as spin and angular momentum in particle physics are now considered metaphorical and have nothing to do with the classical meanings of the words. In particle physics, an electron is considered a zero-dimensional entity with zero volume so it's meaningless to talk about it rotating in any literal sense. Spin is simply one of several quantum numbers that identify particles, and how they interact with other particles, and have nothing to do with any measurable characteristics of macroscopic objects. Julian Schwinger said, "For fundamental properties, we will borrow only names from classical physics".
Spin is described by an SU(2) symmetry. SU(2) is a non-Abelian symmetry. The idea of non-Abelian gauge theories was first suggested by Chen Ning Yang and Robert Mills in 1954. Such theories are called Yang-Mills theories. In 1971 - 1972, Gerard 't Hooft, and Martinus Veltman came up with a renormalizable version of Yang-Mills, for which they received the 1999 Nobel Prize in physics. You can graphically represent spin in the following way. You could imagine drawing a vector in spin space, where it either points up or points down. Thus the two states are spin up or spin down. These are considered two states of one particle.
Then people thought up the idea that you could have a symmetry analogous to spin called isospin, where the two states of a particle are not spin up and spin down, but the proton and neutron. This might seem unusual, but from the point of view of particle physics, the proton and neutron are almost identical. Look at the masses of the proton and neutron.
proton - 938.28 MeV
neutron - 939.57 MeV
Their masses differ by only 0.1%.
The proton has electric charge of +1, and the neutron has electric charge of 0. However, electromagnetism is only 1% as strong as the strong force. Due to these slight differences, the SU(2) symmetry is not exact but it's close.
Isospin is also described by an SU(2) symmetry. You can imagine drawing a vector in isospin space, and if it points up, it's a proton, and if it points down, it's a neutron. These are considered two states of a single particle called a nucleon. It's mathematically the same as spin in that the isospin generators satisfy
[Ij, Ik] = i[epsilon]jklIl
In the fundamental representation, the generators are denoted Ii = (½ )τi where
are the Pauli spin matrices. They act on the proton and neutron states.
The nucleon then forms a doublet.
Other hadrons can also be classified as states in SU(2) multiplets. The less similar the particles, the less the symmetry will hold. The pions are placed in the following states.
where
[pi]± = (-[pi]1 ± I[pi]2)/[squareroot of 2]
[pi]0 = [pi]3