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Free Relativistic Particles and Fields

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Free Relativistic Particles and Fields Hope not without despair, despair not without hope! Seneca (4 BC–65) 4 Free Relativistic Particles and Fields Having learned how the many-particle Schr¨odinger theory can be reformulated as a quantum field theory, we shall now try to find possible field theories for the descrip- tion of relativistic many-particle systems. This will first be done classically. The fields will be quantized in Chapter 7. 4.1 Relativistic Particles The nonrelativistic energy-momentum relation used in the Schr¨odinger theory p2 ε(p)= (4.1) 2M is valid only for massive particles which move much slower than the velocity of light [2] c =2.99792458 1010cm/sec. (4.2) × If particles are accelerated to large velocities close to c this condition is no longer fulfilled. Instead of (4.1), the energy follows the relativistic law ε(p)= c2p2 + c4M 2. (4.3) q In particular, the light particles themselves, the photons, follow this law with the mass M = 0. It will be convenient to replace the energy by the new variable p0 ε(p)/c. (4.4) ≡ Then the relation (29.15) can be expressed as p02 p2 = M 2c2. (4.5) − Thus, energy and momentum of a particle of mass are always such that the four- vector pµ =(p0,pi) (4.6) is situated on the upper hyperboloid with p0 > 0 in a four-dimensional energy- momentum space. This is called the mass shell of the particle of mass M. If the 240 4.1 Relativistic Particles 241 particles are massless, the hyperboloid degenerates into a cone, the so-called light cone. Since a free particle remains free when seen from any rotated, or uniformly moving, coordinate frame, energy and momentum transform in a way that keeps them always on the same mass shell. For a simple rotation of the frame this is obvious. The energy remains the same while the momentum p changes only its direction. For example, p may appear rotated around the z-axis by a transformation ′i i j p = R3(ϕ) jp , (4.7) where R3(ϕ) is the matrix cos ϕ sin ϕ 0 − R3(ϕ) sin ϕ cos ϕ 0 . (4.8) ≡ 0 01 The angle ϕ is defined in such a way that, in the rotated frame, the momenta of the same particles appear rotated in the anticlockwise direction in the xy-plane, i.e., the coordinate axes are rotated clockwise with respect to the original frame. We speak of a passive rotation of the system. The effect is the same as if the observer had remained in the same frame but the experimental apparatus had been rotated in the anticlockwise sense, and with it all particle orbits. The transformations defined in this way are called active transformations. There are two equivalent ways of formulating all invariance principles, one based on the active and one on the passive way. In this text we shall use the passive way. The reader should be aware that different texts use different conventions and the formulas calculated in one cannot always be compared directly with those in the other, but may require changes which fortunately are rather straightforward. For a general rotation by an angle ϕ with an axis pointing in the direction of the 1 unit vector ³ˆ, the transformation has the matrix form ′i i j p = R³ˆ (ϕ) jp . (4.9) We shall also write, with a slightly shorter notation for the rotation matrix, ′i i j p = R j(³) p . (4.10) Explicitly, this transformation reads ′ p = cos ϕ p + sin ϕ (³ˆ p)+ p . (4.11) ⊥ × || Here p||, p⊥ are the projections parallel and orthogonal to the rotation axis ³ˆ: ³ p (p ³ˆ) ˆ, p p p , (4.12) || ≡ · ⊥ ≡ − || 1Hats on vectors in this section denote unit vectors, not Schr¨odinger operators. 242 4 Free Relativistic Particles and Fields respectively. The set of all rotations form a group called the rotation group. Consider now another set of transformations in which the second frame moves with velocity v into the z-direction of the first. In the new frame, the z momentum − of the particle will appear increased. The particle appears boosted in the z-direction with respect to the original observer. The momenta in x- and y-directions are unaffected. Since the total four-momentum still satisfies the mass shell condition (4.3), the combination p02 p32 has to remain invariant. This implies that there must be a hyperbolic transformation− mixing p0 and p3 which may be parametrized by a hyperbolic angle ζ, called rapidity: p′ 0 = cosh ζp0 + sinh ζp3, p′ 3 = sinh ζp0 + cosh ζp3. (4.13) This is called a pure Lorentz transformation. We may write this transformation in a 4 4 -matrix form as × cosh ζ 0 0 sinh ζ µ 0 10 0 p′µ = pν B (ζ)µ pν. (4.14) 0 01 0 ≡ 3 ν sinh ζ 0 0 cosh ζ ν The subscript 3 of B3 indicates that the particle is boosted into the z-direction. A similar matrix can be written down for x and y-directions. In an arbitrary direction ˆ , the matrix elements are cosh ζ ζˆi sinh ζ Bˆ(ζ) B( )= . (4.15) ≡ ζˆi sinh ζ δij + ζˆiζˆj(cosh ζ 1) − The spatial velocity of a particle is given by v ∂ε(p)/∂p. (4.16) ≡ In Schr¨odinger theory this is the velocity of a wave packet. In terms of v v , one defines the Einstein parameter ≡| | 1 γ = cosh ζ. (4.17) ≡ 1 v2/c2 − q With these quantities, we can rewrite (4.15) as γ γvi/c B()= , (4.18) γvi/c δij +(γ 1)vivj/v2 − where (γ 1)vivj/v2 is equal to γ2vivj/c2(γ 1). − − 4.1 Relativistic Particles 243 By combining rotations and boosts, one obtains a 6-parameter manifold of ma- trices ³ ³ Λ(, )= B( )R( ). (4.19) These are called proper Lorentz transformations. For all these, the combination p′02 p′2 = p02 p2 = M 2c2 (4.20) − − is invariant. These matrices form a group, the proper Lorentz group. We can easily see that the Lorentz group allows reaching every momentum pµ on the mass shell µ by applying an appropriate group element to some fixed reference momentum pR. µ For example, if the particle has a mass M we may choose for pR the so-called rest momentum µ pR =(Mc, 0, 0, 0), (4.21) and apply the boost in the pˆ-direction Λ()= B( ), (4.22) with the rapidity given by p0 p cosh ζ = , sinh ζ = | | . (4.23) Mc Mc With this, we can rewrite the general boost matrix (4.15) in the pure momentum form p0/M p pi/M 2c2 B()= | | . (4.24) pi p /M 2c2 δij +ˆpipˆj (p0/M 1) | | − Instead of (4.22), we may use as a boost in the pˆ-direction the more general expressions ³ Λ(p)= B()R( ), (4.25) µ where R(³) is an arbitrary rotation. Also these leave the rest momentum pR invari- ant. In fact, the rotations form the largest subgroup of all proper Lorentz trans- µ formations which leaves the rest momentum pR invariant. It is referred to as the little group or Wigner group of a massive particle. It has an important physical significance since it serves to specify the intrinsic rotational degrees of freedom of the particle. If the particle is at rest it carries no orbital angular momentum. If it happens that its quantum mechanical state remains completely invariant under the little group R, the particle must also have zero intrinsic angular momentum or zero spin. Besides this trivial representation, the little group being a rotation group can 1 3 have representations of any angular momentum s = 2 , 1, 2 ,... In these cases, the state at rest has 2s + 1 components which are linearly recombined with each other upon rotations. 244 4 Free Relativistic Particles and Fields The situation is quite different in the case of massless particles. They move with the speed of light and pµ cannot be brought to rest by a Lorentz transformation from the light cone. There is, however, another standard reference momentum from which one can generate all other momenta on the light cone. It is given by pµ = (1, 0, 0, 1) p , (4.26) R | | with an arbitrary size of the spatial momentum p . It remains invariant under | | a different little group, which is again a three-parameter subgroup of the Lorentz group. The little groups will be discussed in detail in Section 4.15.3. It is useful to write the invariant expression (4.20) as a square of a four-vector pµ formed with the metric 1 1 g = , (4.27) µν − 1 − 1 − namely 2 µ ν p = gµνp p . (4.28) In general, we define a scalar product between any two vectors as pp′ g pµp′ν = p0p′0 pp′. (4.29) ≡ µν − Following Einstein’s summation convention, repeated greek indices are summed from zero to 3 [recall (2.101)]. A space with this scalar product is called Minkowski space. It is useful to introduce the covariant components of any vector vµ as v g vν. (4.30) µ ≡ µν Then the scalar product can also be written as ′ ′µ pp = pµp . (4.31) With this notation, the mass shell properties (4.20) for a particle before and after a Lorentz transformation simply reads p′2 = p2 = M 2c2. (4.32) Note that, apart from the minus signs in the metric (4.27), the mass shell condition p2 = p02 p12 p22 p32 = M 2c2 which is invariant under Lorentz transformations, is − − − 2 2 2 2 completely analogous to the spherical condition p4 + p1 + p2 + p3 = M 2c2 which is invariant under rotations in a four-dimensional euclidean space.
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