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Antiparticles 1 the Klein Gordon Equation Antiparticles PHYS 461 { Fall 2013 The inevitable consequence of merging Quantum Mechanics and Special Relativity is the emergence of antiparticles. Non-relativistic quantum mechanics merely tells us that the evolution of a particle state in Hilbert space obeys a \wavelike" equation that predicts the probability of where and how we might find that particle. But in all cases, there is only one particle in the equations. 1 The Klein Gordon Equation The Schr¨odingerequation is the result of converting the classical conservation of mechanical energy expression to its quantum operator equivalent. Replacing x ! X, ~ @ @ p ! P = i @x , and E ! i~ @t , we obtain p2 @ 2 @2 E = + U(x) =) i (x; t) = − ~ (x; t) + U(X) (x; t) 2m ~@t 2m @x2 The first step toward a relativistic theory of quantum mechanics was to replace the classical energy E with its space-time equivalent, the Lorentz nvariant mass 2 µ 2 2 m = p pµ = E − p This yielded the equation m2 @2 @2 + − (xµ) = 0 ~2 @t2 @x2 As with its classical counterpart, the above expression is also Lorentz invariant, since the derivative terms form the d'Alembertian operator @2 @2 = @µ@ = − µ @t2 @x2 This was the Klein Gordon equation, 2 µ ( + m ) (x ) = 0 whose solutions (also Lorentz invariant!) represent a free particle of mass m moving through space-time with momentum p, µ µ 2 2 2 2 (x ) = C exp(i p xµ=~) = C exp(i(E t − p x )=~) The energy of this particle is E = pm2 + p2... or is it? Any astute observer will note that the energy is actually E = ±pm2 + p2. That is, the Klein-Gordon admits negative-energy solutions! At first, these solutions were dismissed as unphysical. After some thought, how- ever, many began to question whether or not there was unseen wisdom in the solu- tions. Equations don't lie, nor are they embarrassed because they say things that others might find counter-intuitive... 2 The Dirac Sea Around the same time, an alternate view of the negative energy problem was being considered by Paul Dirac. In an attempt to rationalize the problem, Dirac proposed that the negative-energy solutions be accepted as fact. Although the Klein-Gordon equation yielded solutions with no spin (scalar bosons), he argued that what was good for non-spin particles was also good for those with spin. At the time, the electron was known to possess spin, so somehow these should have the opportunity to acquire neg- ative energy. But how could that manifest itself in a world of strictly positive energy?! Borrowing from the theory of atomic orbital structure, Dirac assumed that our universe was full of negative energy electrons, and they completely filled the accessi- ble levels. He dubbed this saturated region of negative energy the Dirac Sea. The only physically accessible energy levels were therefore the positive ones. Since physi- cal process relied on energy differences, this “infinite” background of negative energy was inconsequential to the universe. Periodically, however, energy fluctuations predicted by the uncertainty principle ∆E∆t ∼ ~ could force a negative energy electron to acquire positive energy and pop out of the Dirac Sea, leaving a deficit of one electron. The liberated electron could then occupy a positive-energy state, and appear to us as an electron that materialized out of thin air (or more formally, the vacuum). The resulting \hole" in the sea pro- vided two things: (1) a deficit of negative energy, and (2) a reduction in the overall negative charge of the sea. Figure 1: The Dirac Sea is composed of an infinite number of negative energy states that are completely filled by electrons. Energy fluctuations via the uncertainty principle permit these electrons to climb to positive energy states. The \hole" left behind is interpreted as a positive energy anti-electron. Reinterpreting the above, an observer in the physical world would see (1) a positive energy \anti-electron" with (2) a positive charge. The complete process appears as an electron-antielectron pair (e−e+) created from the vacuum. Dirac's hypothesis had predicted the existence of the antiparticle, several years before it was actually discovered. And what was good for the electron must be good for everything else! 3 The Dirac Equation Dirac sought to formulate a more mathematically-robust description of relativistic particles that included spin. Since the underlying problem of the Klein-Gordon equa- tion was the existence of negative energy solutions (Dirac humoured it with the \sea" theory, but still wasn't convinced), he proceeded to remove them by \taking the square root" of the KG equation. Specifically, he factored the operator portion to give 2 + m −! (iγ@ + m)(iγ@ − m) where the term γ represents a spin-dependent quantity. Furthermore, to incorpo- rate spin, this equation needed to be a multi-component object whose solutions were spinors. This indicated the γ objects needed to be matrices, and moreover needed to be some combination of the Pauli spin matrices. The negative energy solution arose from the (iγ@ +m) term, so this was discarded. What remains is the Dirac Equation, µ µ (ıγ @µ − m) (x ) = 0 µ which of course is Lorentz invariant because γ @µ is the same in every reference frame. A short-hand notation for this (theoretical physicists always have short-hand notations) for this is writing @ @ @ @ γµ@ = γ0 + γ1 + γ2 + γ3 = =@ µ @t @x @y @z pronounced \d-slash".1 A few things to note about the Dirac equation: 1 Note that @µ = (@t; +@x; +@y; +@z), and not @µ = (@t; −@x; −@y; −@z) as we would expect for other four-vectors. This is because the operators are the \reciprocals" of xµ = (t; +x; +y; +z). µ The \downstairs" version of xµ = (t; −x; −y; −z), while it is the \upstairs" version of @ = (@t; −@x; −@y; −@z). • The operator term is a matrix, and the solution (xµ) is a spinor. Specifically, it is a four-component spinor of the form 0 φ+ 1 − B φ C −i pµx (x) = B C e µ (1) @ ξ+ A ξ− The first two terms φ± are a two-component spinor representing the positive- energy spin-up and spin-down solutions. The second two terms ξ± are the negative-energy solutions, a.k.a. the antiparticle solutions. 3.1 More About the Gamma Matrices Dirac's motivation for including spin in his equation necessitates a specific form of the gamma matrices γµ. Since the solutions represent two spinors, the matrices must therefore be a combination of the Pauli spin matrices for each particle. It can be shown that these must have the following block-diagonal form: 0 0 σi γ0 = I ; γi = (2) 0 −I −σi 0 The commutation properties of the spin matrices, [σi; σj] = 2inkσk, directly yield a particularly special property of the gamma matrices: fγµ; γνg = γµγν + γνγµ = 2ηµν That is, the gamma matrices anticommutator is (twice) the underlying space-time metric ηµν! 3.2 Formal Solutions to the Dirac Equation We can write down the explicit form of the particles φ and ξ by re-writing the Dirac equation in operator form by modifying form of the to explicitly include the par- ticle's momentum and energy. Simplifying the form of the solution in Equation 1 to φ µ µ = e−i p xµ = u(p)e−i p xµ ξ and plugging it into the Dirac equation, we find µ e−i p xµ i(−iγ0E + i~γ · ~p)u(p) − mu(p) = 0 Using the explicit structure of the gamma matrices from Equation 2, we get the operator equation µ E − m −~p · ~σ φ e−i p xµ = 0 ~p · ~σ −(E + m) ξ The exponential term can be cancelled out, and we get two equations that mix the states and ξ: (E − m)φ − (~p · ~σ)ξ = 0 (3) (~p · ~σ)φ − (E + m)ξ = 0 (4) Solving each of the above for φ and ξ (i.e. after lots of tedious algebra), we find two solutions 0 q E+m 1 p φ µ 2m −i p xµ + = 2m e (5) @ q E−m A 2m (~p · ~σ)φ 0 q E−m 1 p (~p · ~σ)φ µ 2m +i p xµ − = 2m e (6) @ q E+m A 2m φ The intricacies of these solutions won't be addressed here, but the following take-home points are important: • Equation 5 is the positive energy solution to the Dirac equation +, which µ is evident from the term e−i p xµ . • Equation 6 is the negative energy solution to the Dirac equation −, which µ is evident from the term e+i p xµ . • Both solutions only depends on the two spin components of φ, thanks to Equa- tion 4. That is, whatever particle ξ represents is some kind of \copy" of φ. That is: ξ is the antiparticle of φ. • The coefficients of the solutions are flipped for + and −, implying that the spin properties of the particle are the opposite of the corresponding antiparticle. We'll revisit these solutions when we discuss Feynman diagrams and calculating scat- tering amplitudes. 4 Special Relativity and Causality To close this section, let's consider the geometric consequences of merging quantum mechanics with special relativity. Events in spacetime are denoted by when and where they occur: that is, we associate to an event a spacetime coordinate xµ = (t; x; y; z). µ µ The spacetime interval between any two points xA and xB is 2 µ µ 2 2 2 2 τ = xBxµ B − xAxµ A ≡= (∆t)BA − (∆x)BA − −(∆y)BA − (∆z)BA (7) This quantity is a Lorentz invariant, which means that the value of τ is seen to be the same in any reference frame.
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