User:YohanN7/Consequences of Lorentz invariance

Consequences of Lorentz invariance

The decomposition of the representations under rotations will, when combined with other requirements in applications than well behaved Lorentz transformation properties, lead to restrictions on which vectors in a representation can actually represent states and operators. For instance, a physical elementary particle should have well defined spin that does not change under Lorentz transformations. If the number of independent components of the quantities is less than the dimension (m + 1)(n + 1) of the irrep, then constraints must be imposed on states and the operators operating on the states.

Further constraints may be brought in by demanding a prescribed behavior under parity transformations, i.e. invariance under space inversion represented by the matrix in the full Lorentz group having (1,−1,−1,−1) on the diagonal and 0 elsewhere. Likewise for time reversal transformations represented by (−1,1,1,1). In the physics terminology, a vector quantity in a representation that transforms into minus itself under the parity transormation is a called a pseudo-vector. If it goes into itself it is just a vector. Analogous terminology exists for scalars quantities and tensor quantities. Theories with pseudo-type objects are (perhaps confusingly) also considered invariant in a certain sense. Parity is said to be conserved. An example of a theory lacking parity invariance is the weak interaction. Similar remarks apply to time reversal invariance and the combination of the two inversions, (−1,−1,−1,−1).

Quantum Mechanics

According to standard quantum mechanical rules, a particle with spin j will need a (2j + 1)-dimensional space so that its spin z-component can take on the values j, j−1, ..., −j. A particle with spin j that transforms under the (m,n) representation must therefore be represented by a state vector that remains in one of the rotationally invariant subspaces.

These subspaces do not mix under rotations but they do mix under boosts. An example is given by the vector representation (½,½), which splits into spin j = ½ − ½ = 0 (1-dimensional, e.g. the time component of the electromagnetic vector potential A) and spin j = ½ + ½ = 1 (3-dimensional, e.g. space components A of A) representations. These subspaces don't mix under rotations.

In the application of the theory to quantum mechanics, there are frequently symmetries due to exchange symmetry of identical particles. These result in equivalence relations on the vector space of quantum states. The corresponding quotient space has a natural vector space structure. Any representation of a group or Lie algebra on the original space, for which the kernel of the quotient map is a stable subspace, will descend to the quotient space.

For example, if S is a space of single-particle states, then V = S ⊗ S is a space of 2-particle states. If the particles are bosons, i.e. if u ⊗ v ≈ u ⊗ v for all u,v ∈ S, then the linear subspace H ⊂ V spanned by expressions of the type c(u⊗v − u⊗v) can be seen as physically equivalent to the zero state (the null vector in V). Any tensor product, π_S1 ⊗ π_S2, of representations of S will respect the induced equivalence relation and pass to the quotient space. This exchange symmetry can be described by the permutation group S₂. It acts by permutation of the tensor factors. Equivalently, it acts by permuting indices on tensor components. More complicated exchange symmetries are described by S_n and its subgroups and representations.

Quantum field theory

A few physically reasonable assumptions will have far reaching consequences when combined with Lorentz invariance within quantum field theory (QFT). In ths section, a few basic assumtions of QFT are explicitly outlined.

The Hilbert space

The space of physical states in QFT is an infinite-dimensional Hilbert space that is built up from single-particle states using tensor products and direct sums. By the usual rules of tensor products, a basis for the one-particle states will yield a basis for any tensor product of 1-particle Hilbert spaces. A similar comment applies to taking direct sums. Every state in the Hilbert space is a superposition of multi-particle states. See the article on Fock space for details.

A typical element of Hilbert space will look like

$A|\alpha _{1}\rangle \otimes |\beta _{1}\rangle \otimes \cdots \otimes |\gamma _{1}\rangle \oplus \cdots \oplus B|\eta _{2}\rangle \otimes |\zeta _{2}\rangle \otimes \cdots \otimes |\theta _{n}\rangle$,

where α_i, β_i, etc. are complete sets of quantum numbers, |α_i〉, |β_i〉, etc. are single particle states, and A,B are constants. Not all of these states are particularly meaningful. The meaningful states will usually exhibit certain exchange symmetries and are subject to normalization in most computations. More general states are given by integrals, most commonly over momentum of multi-particle states with definite momenta in each factor. An example of this type is given in the following sections.

Linear operators Hilbert space

The construction allows a particularly useful basis for the set of linear operators on the space. The creation and annihilation operators are specified by defining their action on the multi-particle states. As their names indicate, they take n-particle states to (n + 1)-particle states and (n − 1)-particle states respectively. It might be noted that the existence of these operators has nothing to do with whether particles can actually be created or destroyed. The effect of the creation operator on a multi-particle state is defined by

$a(q)^{\dagger }|q_{1}\rangle \otimes |q_{2}\rangle \otimes \cdots \otimes |q_{n}\rangle =|q\rangle \otimes |q_{1}\rangle \otimes |q_{2}\rangle \otimes \cdots \otimes |q_{n}\rangle .$

In particlular, if |VAC> denotes the vacuum, then

$a^{\dagger }(q_{1})a^{\dagger }(q_{2})\cdots a^{\dagger }(q_{n})|VAC\rangle =|q_{1}\rangle \otimes |q_{2}\rangle \otimes \cdots \otimes |q_{n}\rangle .$

Here q denotes the complete set of quantum observables {p,σ,n} where n is the particle type, p 3-momentum, and σ is the spin z-component. If there are more discrete quantum numbers, they are assumed to be included in the σ-label.

A more intricate state is given by a possibly bound particle antiparticle pair, e.g. positronium, given by

$\Phi =\sum _{\sigma \sigma ^{\prime }}\int \int \chi ({\textbf {p}},\sigma ,{\textbf {p}}^{\prime },\sigma ^{\prime })a^{\dagger }({\textbf {p}},\sigma )a^{c\dagger }({\textbf {p}}^{\prime },\sigma ^{\prime })|VAC\rangle d^{3}{\textbf {p}}d^{3}{\textbf {p}}^{\prime }$

where a^† creates a particle and a^†c its antiparticle, and χ is the wave function.

The annihilation operator is defined to be the adjoint of a,

$a(q)=(a(q)^{\dagger })^{\dagger }.$

Its effect on an n-particle state is slightly more complicated due to the possible exchange symmetries described below. It is in any case a linear combination (n terms) of (n − 1)-particle state.

The creation and annihilation operators usually obey relations among themselves. This is typically expressed by commutator or anticommutator relations between them. Physically, these relations origin in various exchange symmetries between states. On Hilbert space they induce equivalence relations resulting in subspaces representing the same physical state. The resulting quotient space represents, up to normalization, the unique physical state. One consequence of this is that the effect of the creation operator a^†(q) of a fermionic particle on a state-vector |α〉 where the state q is occupied is to destroy the state, a^†(q)|α〉 = 0.

Any linear operator on Hilbert space can be expressed in terms of creation and annihilation operators. The expression is a polynomial the a, a^† with momentum-dependent coefficients integrated over all momenta.

As a consequence of the (anti-) commutation relations for bosonic and fermionic fields, the Hamiltonian takes the simple form

$H=\int a(q)^{\dagger }a(q)E(q)dq,E(\mathbf {p} ,\sigma ,n)={\sqrt {\mathbf {p} ^{2}+m_{n}^{2}}}$

Here, dq is a shorthand for summing over particle types and discrete labels, and integrating over the continuous labels (momenta).

Transformation of single-particle states

The single particle states are assumed to transform under some, not necessarily irreducible, representation of the Lorentz group. To say this again, a state representing a physical free particle is assumed to have definite Lorentz transformation properties.

Free one-particle states can be characterized by a set of labels {p, σ, ...} where p is linear momentum, σ is the spin z-component or helicity for massless particles, and the ellipsis denote other discrete labels. Under a Lorentz transformation of the space–time variables (t,x,y,z) ↦ (t′,x′,y′,z′) a one particle state |p,σ,...〉 vector (bra-ket notation) will be affected by a unitary or antiunitary transformation |p,σ,...〉 ↦ |p′,σ_−j′,...〉 + |p′,σ_−j+1′,...〉 ... + |p′,σ_j′,...〉 of Hilbert space. Wigner's theorem asserts and proves the existence of such a transformation.

With the choice of parameters as above, p transforms under the 4-vector representation (½,½). Thus for a Lorentz transformation Λ in the standard 4-vector representation (½,½), p′ = Λp (matrix multiplication). The σ-label will transform under some finite-dimensional representation. Considered as a column vector σ transforms as σ = C(Λ,p)σ, where C is a matrix. The complete expression for a free massive single-particle state reads

$U(\Lambda )|p,\sigma \rangle ={\sqrt {\frac {(\Lambda p)^{0}}{p^{0}}}}\sum _{\sigma ^{\prime }}D_{\sigma ^{\prime }\sigma }^{(j)}(W(\Lambda ,p))|\Lambda p,\sigma ^{\prime }\rangle$,

where W(Λ,p) ⊂ SO(3) is the Wigner rotation corresponding to Λ and p. The Wigner rotation is a consistently chosen rotation for a Lorentz transormation taking a massive particle at rest to momentum p. The matrix D is the (2j + 1)-dimensional representation of the rotation group SO(3).

In a (only slightly) less abstract setting, the ket |p,σ〉 may be represented by functions of space-time (with p as a parameter) as entries in a (2j + 1)-dimensional column vector. In this case the functions will be e^ipx. Other sets of parameters are also possible. One can also, for instance, use the set {p_r,j(j+1),σ} where p_r is a continuous index representing "radial momentum", and j(j + 1) is total angular momentum. In this case, the corresponding functions are built up from spherical harmonics and spherical Bessel functions. These infinite sets of functions must transform among themselves under the infinite-dimensional representations of the Lorentz group.

The set {e^ipx} does constitute representation of the Lorentz group using the rule D(Λ)e^ip·x ↦ e^{iΛ−1p·x′} where D(Λ) is an infinite-dimensional representation on function space of Λ taking x to x′. It is not irreducible however.

Transformation of multi-particle states

The transformation properties of multi-particle states follow from the properties of the single-particle states under formation of direct sums and tensor products of representations. The properties of more complicated states (e.g. coherent states) follow by linearity.

Transformation of linear operators

The transformation properties the creation and annihilation operators follow too using the representations induced on End(H) and hence the transformation properties of all operators once they are expressed in terms of creation and annihilation operators. The transformation rule for the creation operator is

$U(\Lambda )a^{\dagger }(p,\sigma ,n)U^{-1}(\Lambda )=\sum _{\sigma ^{\prime }}D_{\sigma ^{\prime }\sigma }(W(\Lambda ,p))a^{\dagger }(p_{\Lambda },\sigma ^{\prime },n).$

The behavior of creation and annihilation operators under Lorentz transformations restricts the form both of the free quantum fields and their interactions. A few consequences for free fields will be outlined below.

The S-matrix

The S-matrix is unitary and assumed to be Lorentz invariant. The first condition follows from its (rigorous) definition. It is a "matrix" connecting two complete sets of basis vectors for Hilbert space, that of the "in states" and that of the "out states".

The unitarity simply says that probability amplitudes S_βα = 〈β|α〉 for processes α → β are the same as those for〈U(Λ)β|U(Λ)α〉. The U(Λ) are the unitary operators on Hilbert space corresponding to the Lorentz transformation Λ. When this is written out explicitly (observing that it holds for all in- and out-states) one obtains a definition of Lorentz invariance of the S-matrix. The precise equation expressing Lorentz invariance of the S-Matrix is rather involved.. In principle, this relation can be expressed in terms of one-particle states and creation and annahilation operators, and their respective known Lorentz transformation properties.

The S-matrix will be Lorentz invariant if the interaction V can be written as

$V=\int {\mathcal {H}}(\mathbf {x} ,t)d^{3}x,$

and the Hamiltonian density transforms as

$U_{0}(\Lambda ,a){\mathcal {H}}(x)U_{0}^{-1}(\Lambda ,a)={\mathcal {H}}(\Lambda x+a),$

and, in addition, the causality condition below is satisfied. The Hamiltonian density is in general a polynomial (with constant coefficients) in the creation and annihilation fields.

Quantum fields

Quantum fields are expressed as linear combinations,

$\psi =\kappa \psi ^{+}+\lambda \psi ^{-}$

of annihilation fields and creation fields,

$\psi _{l}^{+}=\sum \int u_{l}(x;p,\sigma ,n)a(p,\sigma ,n)d^{3}p,\qquad \psi _{l}^{-}=\sum \int v_{l}(x;p,\sigma ,n)a^{\dagger }(p,\sigma ,n)d^{3}p.$

Here, the a* is the creation operator, tacking on a single particle of type n with momentum p and spin z-component σ to any state (ignoring exchange symmetries). The annihilation operator a* is its adjoint. The index l runs over all considered particle types and also over all irreducible representations as well as components of these representations.

The requirement of Lorentz invariance of the S-matrix, when applied to the fields, using known properties of the creation and annihilation operators, leads to the equations

$\sum _{\bar {\sigma }}u_{\bar {l}}(\Lambda x+b;{\textbf {p}}_{\Lambda },{\bar {\sigma }},n)D_{{\bar {\sigma }}\sigma }^{(j_{n})}(W(\Lambda ,p))={\sqrt {\frac {p^{0}}{(\Lambda p)^{0}}}}\sum _{l}D_{{\bar {l}}l}(\Lambda )e^{(i(\Lambda p)\cdot b)}u_{l}(x;{\textbf {p}},\sigma ,n),$

$\sum _{\bar {\sigma }}v_{\bar {l}}(\Lambda x+b;{\textbf {p}}_{\Lambda },{\bar {\sigma }},n)D_{{\bar {\sigma }}\sigma }^{(j_{n})*}(W(\Lambda ,p))={\sqrt {\frac {p^{0}}{(\Lambda p)^{0}}}}\sum _{l}D_{{\bar {l}}l}(\Lambda )e^{(-i(\Lambda p)\cdot b)}v_{l}(x;{\textbf {p}},\sigma ,n).$

The u and v are referred to as coefficient functions. In the sequel it will be seen that these functions, and hence the field operator, will satisfy certain differential equations. In the parametrization using p it is seen by considering translations (the full Poincaré group is considered) that

$u_{l}(x;p,\sigma )=(2\pi )^{-3/2}e^{ipx}u_{l}(p,\sigma )$ and

$v_{l}(x;p,\sigma )=(2\pi )^{-3/2}e^{-ipx}v_{l}(p,\sigma )$,

where the species index n have been dropped.

For zero momemtum, by considering rotations and infinitesimal rotations in turn, one obtains the relations

${\begin{aligned}\sum _{\bar {\sigma }}u_{\bar {l}}(0,{\bar {\sigma }})\mathbf {J} _{{\bar {\sigma }}\sigma }^{(j)}&=\sum _{\bar {l}}\mathbf {\mathcal {J}} _{{\bar {l}}l}u_{l}(0,\sigma ),&\sum _{\bar {\sigma }}u_{\bar {l}}(0,{\bar {\sigma }})D_{{\bar {\sigma }}\sigma }^{(j)}(R)=\sum _{\bar {l}}D_{{\bar {l}}l}(R)u_{l}(0,\sigma )\\\sum _{\bar {\sigma }}v_{\bar {l}}(0,{\bar {\sigma }})\mathbf {J} _{{\bar {\sigma }}\sigma }^{(j)*}&=-\sum _{\bar {l}}\mathbf {\mathcal {J}} _{{\bar {l}}l}u_{l}(0,\sigma ),&\sum _{\bar {\sigma }}v_{\bar {l}}(0,{\bar {\sigma }})D_{{\bar {\sigma }}\sigma }^{(j)*}(R)=\sum _{\bar {l}}{D}_{{\bar {l}}l}(R)u_{l}(0,\sigma )\end{aligned}}$

for the Lie algebra representations (left) and the group. In these equation, the J are spin matrices for spin j, and the MATHCAL J is some, not necessarily irreducible, representation of so(3;1). The D are representations of the Lorentz group, while the D^j are representations of SO(3).

The behavior of u and v is governed strongly by which (m,n) irrep under which the fields transform. One first considers how the fields must appear at zero momentum, p = 0 (massive particles only). The coefficient functions have (m + 1)(n + 1) components, but only (2j + 1) of those can be independent (corresponding to the allowed values for σ). It is, in principle, easy to find u(0) and v(0) if (m + 1)(n + 1) = (2j + 1). Additional assumptions, like parity invariance are taken into account at this point. If (m + 1)(n + 1) ≠ (2j + 1), then further constraints must be imposed.

With knowledge of u_l(0,σ) and v_l(0,σ) the appearance at finite momenta p can be found by applying a standard (m,n) transformation corresponding to a specific Λ(p) taking (0,0,0) to p to (the vectors, spinors, tensors or spinor-tensors) u and v respectively. These standard are given by

$u_{l}({\textbf {p}},\sigma )=(m/q_{0})^{\tfrac {1}{2}}\sum _{\bar {l}}D_{l,{\bar {l}}}(L(p))u_{\bar {l}}({\textbf {0}},\sigma ),\qquad v_{l}({\textbf {p}},\sigma )=(m/q_{0})^{\tfrac {1}{2}}\sum _{\bar {l}}D_{l,{\bar {l}}}(L(p))v_{\bar {l}}({\textbf {0}},\sigma ),$

where L is a standard Lorentz boost taking zero momentum to q, and D is its representation.

Causality

The principle of causality is assumed to hold. The latter can be expressed more technically by assuming the slightly weaker cluster decomposition principle. In this setting one finds that the Hamiltonian density, and hence free field operators must commute at spacelike distances by using the known transformation properties of the creation and annihilation operators. If not, the cluster decomposition may be violated meaning, in principle, that experiments made at CERN can interfere with experiments at Fermilab or elsewhere in the universe. Mathematically the causality principle now reads

$[{\mathcal {H}}(x),{\mathcal {H}}(y)]=0\Rightarrow [\psi _{l}(x),\psi _{l^{\prime }}^{\dagger }(y)]=0$

for (x−y) spacelike, where † denotes the adjoint, and l, l′ are component indices of the field operator.

Free field equations and gauge principles

The commutator equation leads to free field equations for the field operators. The basic example is that all components of all massive quantum fields satisfy the free Klein–Gordon equation.

For a spin ½ particle with mass m in the (½,0) ⊕ (0,½) representation, the added assumption of parity invariance under the full Lorentz group the causality principle leads to the free field Dirac equation. Starting with the MASTER EQUATION, using the Pauli spin matrices for spin ½, an application of Schur's lemma leads to the (most) general ansatz

$u(0,{\tfrac {1}{2}})={\begin{bmatrix}c_{+}\\0\\c_{-}\\0\end{bmatrix}},u(0,-{\tfrac {1}{2}})={\begin{bmatrix}0\\c_{+}\\0\\c_{-}\end{bmatrix}},v(0,{\tfrac {1}{2}})={\begin{bmatrix}0\\d_{+}\\0\\d_{-}\end{bmatrix}},v(0,-{\tfrac {1}{2}})=-{\begin{bmatrix}d_{+}\\0\\d_{-}\\0\end{bmatrix}}$

for the coefficient functions at 0 momenta. The choice of an overall scale and parity invariance fixes two of the unknown parameters in the ansatz. The commutator equation explicitly reads

$[\psi _{l}(x),\psi _{\bar {l}}^{\dagger }]=(2\pi )^{-3}\int [|\kappa |^{2}N_{l{\bar {l}}}({\textbf {p}})e^{ip\cdot (x-y)}-|\lambda |^{2}M_{l{\bar {l}}}({\textbf {p}})e^{-ip\cdot (x-y)}]d^{3}p=0,$ where

$N_{l{\bar {l}}}({\textbf {p}})=\sum _{\sigma }u_{l}({\textbf {p}},\sigma )u_{\bar {l}}^{*}({\textbf {p}},\sigma ),M_{l{\bar {l}}}({\textbf {p}})=\sum _{\sigma }v_{l}({\textbf {p}},\sigma )v_{\bar {l}}^{*}({\textbf {p}},\sigma ).$

The equation fixes the last unknowns in the zero momentum coefficient functions and further implies that thy satisfy (ip^μγ_μ + m)u(p,σ)_l = 0 and (−ip^μγ_μ + m)u(p,σ)_l = 0 respectively. These are the momentum space versions of the Dirac equation and its adjoint in its original form. The index l runs over the 4 components of the Dirac field. The γ^μ are the Gamma matrices, also called the Dirac matrices, of dimension 4. For the field operator one obtains

$(\gamma ^{\mu }\partial _{\mu }-m)\psi =0.$

The appearance of the partial derivative is a consequence of properties (p ↔ id⁄dx) of the Fourier transform. This free field equations is obeyed by all free massive spin-½ particles having party invariance in the (½,0) ⊕ (0,½) representation.

The approach used here should be contrasted with the method of canonical quantization starting with a Lagrangian density for the free field and postulating canonical equal time commutation relations for the field operator. The equations obtained are precisely the Heisenberg equations of motion for the field operator. The corresponding Schrödinger picture equations can be obtained by standard means. These equations may or may not have formal similarity with the corresponding Heisenberg equations. In the case of the Klein–Gordon field, the Dirac field, the electoromagnetic field (below), and the Proca field, the equations of motion are formally identical.

Similar considerations and the (1,0) ⊕ (0,1) representation lead to the free field Maxwell equation. The Maxwell field tensor, f^μν resides in the (1,0) ⊕ (0,1) representation, which is 16-dimensional, but it has only 6 independent components. These are related by the free field Maxwell equation.

Principles of gauge invariance arise in this way too. When considering massless fields x^μ of spin 1 in the 4-vector representation (½,½), one finds that such fields will not be 4-vectors in general. There are degrees of freedom that does not correspond to physical degrees of freedom. These fields are nonetheless useful because Lorentz invariant quantities can be constructed from them. One example is the Maxwell field which is given by

$f_{\mu \nu }=\partial _{\mu }a_{\nu }-\partial _{\nu }a_{\mu }.$

Lorentz scalars (i.e. (0,0) representations) can be formed by contraction. The quantity A^μΨ_μ, where Ψ is the electron–positron field, is an ingredient in the Lagrangian in quantum electrodynamics (QED) representing the interaction between electrons and photons.

Other consequences

A couple more profound consequences of Lorentz invariance in QFT include the following.

• The existence of antiparticles
• The CPT theorem
• The spin-statistics theorem