Lorentz and Poincaré Group and its representations
2.1. General considerations – Quantum Mechanics and Symmetries
First, we will study how symmetries appear in a quantum setting, just like Lorentz invariance.
The quantum field theory is based on quantum mechanics that was invented in 1925 – 1926 by Heisenberg, Schrödinger, Pauli, Born, and others. This theory is been used since in nuclear, atomic, molecular and condensed matter physics.
The physical states are represented in Hilbert space by rays instead of vectors. A Hilbert space is a vector space that has natural inner product, or dot product, providing a distance function. Under this distance function it becomes a complete metric space and, thus, is an example of what mathematicians call complete inner product space.
It is a kind of complex vector space. If ? and ? are vectors in the space, they are often called ‘state – vectors’, then so is ??+??, for arbitrary complex numbers ?,?. It has a norm*: for any pair of vectors there is a complex number?,?, such that
The norm ?,? satisfies a positivity condition:?,??0, and become 0 if and only if ?=0.
By rays we mean classes of equivalence of vectors (normalized) which differ one from another by a phase vector. In fact, a ray is a set of normalized vectors (in other words ?,?=1) with ? and ?’ belonging to the same ray if ?’=??, where ? is an arbitrary complex number with ?=1.
In quantum mechanics, observables are represented by linear Hermitian operators. Also, they act a lot like random variables. In fact, these are mappings ??A? of Hilbert space into itself, linear in the sense that
and satisfying the reality condition A?=A, where for any linear operator A the adjoint A? is defined by
Where the operator A? conjugate to a linear operator A.
We can suppose that the continuity of A? is a function of ?. A state represented by a ray R has a definite value ? for the observable represented by an operator A if vectors ? belonging to this ray are eigenvectors (special set of vectors associated with linear system of equations that are sometimes also known as characteristic vectors, proper vectors, or latent vectors, in other words a matrix equation) of A with eigenvalue ?:
A?=?? for ? in R(2.1.6.)
From the definition below it follows that eigenvalues of Hermitian operators are real, so ? is a real eigenvalue. After another theorem, different state vectors like ?1 and ?2 on a given Hermitian operator A has different eigenvalues, ?1 and ?2, are orthogonal to each other.
The concept of Hermitian operator is extended in quantum mechanics to operators that don’t need second-order differential.
If a system is in a state represented by a ray R, and an experiment is done to test whether it is in any one of the different states represented by mutually orthogonal rays R1,R2, …then the probability of finding it in the state represented by Rn is
where ? and ?n are any vectors belonging to rays R and Rn, respectively. A pair of rays are orthogonal if the state-vectors from the two rays have vanishing scalar products. Another elementary theorem gives a total probability unity:
if the state-vectors ?n form a complete set.
We can define a symmetry transformation a change when the outcome of measurement, the results of possible experiments does not change. Symmetry transformation of a given physical system forms a group. We can take an observer O that sees a system in a state represented by a ray R or R1 …, then a correspondent observer O’ who looks at the same system can observe it in different state, represented by a ray R’ or R1′ …, but these two observers must have the same probabilities
where R, Rn … are the rays representing states of the original system and R’, Rn’ … are the rays representing states of the transformed system, in other words the states of the original system but described by another observer.
In the Hilbert space symmetry transformations are represented by some operators acting on vectors. An essential theorem by Eugen Wigner says that any symmetry transformation R?R’of rays on Hilbert space is represented by a linear and unitary operator U
where ? is in ray R and U? is in the ray R’,
or antilinear and antiunitary operator
Defining the Hermitian conjugation A? of the antilinear operator A by
we obtain, that for both, linear and antilinear operators, the unitarity or antiunitarity conditions take the form
Regularly there is a trivial symmetry transformation which is represented by the unit operator U=1, which is unitary and linear. It follows, that any symmetry that can be made trivial by changing some of the parameters must be represented by a linear unitary operator U instead one operator that is antilinear and antiunitary. For example rotation, translation or Lorentz transformation, these are represented by linear and unitary operators, but the time reversal transformation symmetry is represented by the antilinear operator.
We can say that a set of symmetry transformations is a group because it has certain properties that define them. In other words the transformation must be reflected in the properties of the operators representing them.
So if T1:Rn?Rn’, T1 is a transformation that takes rays from Rn into Rn’, and
T2:Rn’?Rn” where T2 is a transformation that takes rays from Rn’ into Rn”, then the result of operating both transformations is another symmetry transformation, which we can write
A T symmetry transformation has an inverse, which we can write T-1. The inverse of the T symmetry transformation takes rays from Rn’ into Rn, and there exists an identity transformation which leaves rays unchanged, T=1.
Therefore the operator U(T2T1) should transform vectors from Rn into vectors from Rn”. But since operators act on vectors instead on rays, one cannot exclude that
We can see that if T1 takes rays from Rn into Rn’, then alternating on a vector ?n in the ray Rn, UT1 must generate a vector UT1?n in the ray Rn’, and in the same way in case of T2 takes rays from Rn’ into Rn”, then alternating on a vector UT1?n must generate a vector UT2UT1?n in the ray Rn”. Because UT2T1?n is in the same ray, so the vectors can differ only by a phase ?nT2,T1.
The linearity or antilinearity of U(T) describes that these phases are independent of the state ?n.
In general the composition law of symmetry operators has the following form:
If ?=0 we can say that U(T) provide a representation of the group of symmetry transformations. For ?T2,T1 that can’t be absorbed by a redefinition of the operators U(T), we speak about a projective representation, or it can also be called an ‘up to a phase’ representation.
We can avoid the projective representations by enlarging the symmetry group. In this case there will be no changing in its physical implications.
In physics exists, an important group, called connected Lie group. These groups are groups with T(?) transformations, which are described by a set of real parameters, like ?a. In this group each piece is connected to the identity by a path.
So the group multiplication law takes the following form
where fa?,? is a function of the ?s and ?s. If we take ?a=0 as the coordinates of the identity, we will have
The transformations of continuous groups we must represent on the physical Hilbert space with unitary operators U(T?), not with antiunitary operators. For Lie groups the operators can be represented in a finite neighborhood of the identity by a power series
where ta,tbc=tcb, … are Hermitian operators independent of the ?s. Suppose that the UT? form an ordinary, in other word not projective representation of this group of transformations
2.2. Characterization of the Poincaré group
In this section we will work with the Lorentz Transformations.
Einstein’s principle of relativity affirms the equivalence of certain ‘inertial’ frames of reference. It is remarkable from the Galilean principle of relativity, accepted by Newtonian mechanics, by the transformation connecting coordinate systems in different inertial frames. If x? are the coordinates in one inertial frame, with x1,x2,x3 Cartesian space coordinates, and x0=t a time coordinate, the speed of light being set equal to unity, then in any other inertial frame, the coordinates x’? must satisfy
????x’??x? ?x’??x?=??? (2.2.2.)
In the equation above, ??? is the diagonal matrix, with elements
This is the Minkowski metric, is a tensor whose elements are defined by the following matrix
Each coordinate transformation x??x’? that accomplish Eq.(2.2.2.) is linear3ax’?=? ?? x?+a?(2.2.4.)
where ? ? ? is a constant matrix and a? are arbitrary constants which satisfies the following Lorentz conditions
??? ? ? ?? ??=???(2.2.5.)
Any transformation ? that satisfies the Eq. (2.2.5.) is a Lorentz transformation. In some problems it is more effective to write the Lorentz transformation condition in another way. The diagonal matrix ??? has an inverse, which is written ???. The inverse’s components are the same like for the original diagonal matrix
In case of multiplying Eq.(2.2.5.) with ??? ? ? ? we will obtain
??? ? ? ?? ? ?? ?? ???=? ??=??? ??? ? ? ?(2.2.7.)
And if we multiply Eq. (2.2.7.) with ??? ? ? ? matrix inverse, we will get
? ? ?? ?? ???=???(2.2.8.)
All of these transformations form a group. We saw at Eq. (2.2.4) representing a first Lorentz transformation, and then a second Lorentz transformation x’ ??x”?, with
x”?=? ??x’?+a?=? ??? ?? x ?+a?+a?(2.2.9.)
and if we use the Lorentz transformation x??x”?, the effect will be the same with
x”?=? ??? ??x?+? ??a?+a?(2.2.10.)
In case if ? ?? and ? ?? both satisfy the Lorentz condition, so does their product ? ??? ?? and so we have the composition rule for these transformations:
The identity transformation is T1,0. To prove this we use the Eq. (2.2.11.) and we will get
The inverse transformation for T?,a will be T?-1,-?-1a. To prove that we use the same equation as for the identity
For every transformation T corresponds an individual linear operator that acts on vectors in Hilbert space such that it takes ??U?,a? and satisfy the composition rule
with identity U1,0 and inverse elements U?-1,-?-1a.
The entire group of specific transformation is known as the inhomogeneous Lorentz group or Poincaré group. The Poincaré group has two important subgroups. The first one is the group of transformations with a?=0. These are known as the homogeneous Lorentz group or just Lorentz group. Utilizing the Lorentz condition (2.2.5.), the property of the product of determinant and the fact that det?=1 we will have
For ? ? ? and ? ? ? we have
?? 00=? 00? 00+? 10? 0 1+? 20? 02+? 30? 0 3(2.2.16.)
In accordance with eq. (2.2.5.) and eq. (2.2.8.) we have
???? 0?? 0?=?00=+1?? 002=1+? 0i? 0i=1+? i0? i0(2.2.17.)
We can observe that ? 002?1 so that either ? 00?+1 or ? 00?-1. Those transformations who, has ? 00?+1 form a subgroup. According to eq. (2.2.17.), the length of the 3-vector ? 0 1,? 02,? 0 3 is ? 002-1. Item the 3-vector ? 0 1,? 02,? 0 3 has length ? 002-1 and so the scalar product of the two 3-vectors is bounded, by
? 10? 0 1+? 20? 02+? 30? 0 3?? 002-1? 002-1?
?? 00-? 00? 00?? 002-1? 002-1?
-? 002-1? 002-1??? 00-? 00? 00?(2.2.19.)
?? 00?? 00? 00-? 002-1? 002-1?1
The identity appears when ?=1 and to have continuity we need to take ? 002?1 (you can see it in Figure 2.2.1.) and det?=1. If we want to obtain a Lorentz transformation from the identity by a continuous change of parameters they must have det? and ? 00 the same sign as the identity, because we can’t jump from det?=+1 to det?=-1, or from ? 00?+1 to ? 00?-1. Therefore it forms a subgroup of Lorentz transformations known as the popper orthochronous Lorentz group.
Figure 2.2.1.: Allowed region
2.3. Natural representation of the Lorentz Group
In this division we will consider the natural representation of the Lorentz group L. The elements L?L act on 4-dimensional vectors of position- and time-coordinates. This will be noted as follows
where x0=t define the time coordinate and x1,x2,x3=x 3-dimensional vector define the space coordinates. All the components of x? have the same dimension that is the length.
The Lorentz transformations were formulated in a space, called Minkowski space, in which the time component was chosen an imaginary number and the space component was chosen a real one. For this plain space we use the metric tensor
The scalar product is given by
x?y?xTg y=x0y0-x ?y=g??x?y?=x?y?(2.3.3.)
It is more favorable to use the index notation for explicit instances of ?, where upper and lower indices are summed over. The x’=?x Lorentz transformations leave the scalar product invariant:
?x??y=x?y ? xT?Tg?y=xTgy? ?Tg?=g(2.3.4.)
Using the components this condition takes the form
g??=g??? ??? ??(2.3.5.)
Because the metric tensor is symmetric, this gives 10 restrictions; the Lorentz transformation ? is a 4 × 4 matrix, so it has 16 parameters and it depends on 16-10=6 independent parameters. If we write an insignificant transformation as ? ??=? ??+? ??+…, then it follows from eq. (2.3.5) that ???=-??? must be completely antisymmetric.
The Lorentz group O3,1 is the orthogonal group Om,n, where the transformation of a space with coordinates y1…yn,x1…xm that leave the quadratic form y12+…+yn2-x12+…+xm2 invariant.
The group axioms are fulfilled; there exist a unit element ?=1, and each ? has an inverse element because it is invertible:
?Tg?=g ? det?2=1 ?det?=±1(2.3.6.)
Eq. (2.3.5.) involve that
g?? ? 0?? 0?=? 002-k? 0k2=1 ? ? 002?1(2.3.7.)
The Lorentz group has four disconnected components, depending of the signs of det? and ? 00. The subgroup with det?=1 and ? 00?1 is named the proper orthochronous Lorentz group SO3,1?. It keeps the direction of time and parity and contains the identity matrix. The other three branches can be constructed from a given ??SO3,1? mixed with a space and time reflection:
SO3,1?× spatial reflections: ? 00?1, det?=-1SO3,1?× time reversal: ? 00?-1, det?=-1SO3,1?× space-time reflection: ? 00?-1, det?=1In Minkowski space the Lorentz transformations keep the norm x2=x?x that is positive for spacelike vectors, positive for timelike vectors or zero for lightlike vectors.
As a matter of fact the Lorentz transformations L describe the relationship between space-time coordinates x? of two reference frames which move relative to each other with uniform fixed velocity ? and which might be reoriented relative to each other by a rotation around a common origin. x? stays for the coordinates in one reference frame and x’? stays for the coordinates in the other reference frame. So the Lorentz transformations constitute a linear transformation which we can write
The Lorentz transformation is represented by a 4×4-matrix with the elements L ??. The first index of the matrix is denoted by ? and the second index is denoted by ?. There exist four possibilities for the positioning of the indices ?,?=0,1,2,3:
4-vector: x?,x?:4×4 tensor: ? ??,?? ?,???,???(2.3.9.)
The Lorentz transformations are non-singular 4×4-matrices with real coefficients, in other word L?GL(4,R). These transformations form the subgroup of all matrices. Connecting condition x?g??x?=x’?g??x’? and eq. (2.3.8.) leads us to
L ? ?g??L ??x?x?=g??x?x?(2.3.10.)
Because this holds for any x? it must be true
L ? ?g??L ??=g??(2.3.11.)
The previous condition specifies the key property of Lorentz transformations.
Now we will classify the elements of L=O(3,1). First we consider the value of detL. Using detAB=detAdetB and detAT=detA generate detL2=1 or detL=±1.We can classify Lorentz transformations conform the value of the determinant into two distinct classes.
Another class property can we deduce from
where g=g??, we can it apply in eq. (2.3.11.). If we take the case when ?=0, ?=0 we will obtain
L 002-L 012-L 022-L 032=1(2.3.13.)
and while L 012+L 022+L 032?0 it holds L 002?1, so we can deduce that
L 00?1 or L 00?-1(2.3.14.)
From this conclusion we can distinguish two other distinct classes of Lorentz transformations.
With these preliminary observations, the set of Lorentz transformations L is given as the union of disjunct sets
These sets are defined as follows
L+?=L, L?O3,1,detL=1, L 00?1 (2.3.16.)
L+?=L, L?O3,1,detL=1, L 00?-1 (2.3.17.)
L-?=L, L?O3,1,detL=-1, L 00?1 (2.3.18.)
L-?=L, L?O3,1,detL=-1, L 00?-1 (2.3.19.)
From these four L+? is a subgroup of SO(3,1) and it’s called the subgroup of proper, orthochronous Lorentz transformations. These are continuously connected to the identity I, in other words these transformations can be parametrized such that a continuous variation of the parameters connects any element of L+? with I.
A field is any set of elements that satisfies the field axioms for both addition and multiplication and is commutative division algebra. Because the identity condition is generally required to be different for addition and multiplication, every field must have at least two elements. It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements are equivalent to the field of complex numbers.
A field is a number of amounts that is a function of space-time point x?=t,x. A scalar field is a scalar quantity that is a function of space time, a vector quantity that is a function of space time is called a vector field and other similar things.
The Lorentz transformation properties and the rank of a field are defined the same way as before, provided that the quantities are evaluated at the same event point before and after a Lorentz transformation, namely
Scalar field: ?’x’=?x(4.1.)
Vector field: ?’?x’=? ????(4.2.)
Tensor field: T’??x’=??????T??x(4.3.)
where x and x’ are connected by
4.1. Free Fields
An S-matrix can be a Lorentz-invariant if the interaction can be written as
where H is a scalar, so theoretically
The scalar H satisfies further condition:
Hx,Hx’=0 for x-x’2?0(4.1.3.)
To satisfy the cluster decomposition principle we must construct the Hx out of creation and annihilation operators. But there appears a problem, bellow Lorentz transformations each this kind of operator is multiplied by a matrix that depends on the momentum carried by that operator. To fix this problem and to couple this kind of operators we must build Hx out of fields: annihilation fields ?l+x and creation fields ?l-x?l+x=?nd3pulx;p,?,nap,?,n(4.1.4.)
The coefficients ulx;p,?,n and ?lx;p,?,n are chosen in the way to under the Lorentz transformations for each field is multiplied with a position-independent matrix:
In principle we have distinct transformation matrices D± for the annihilation and creation fields, it might be that these matrices we be the same. Applying another Lorentz transformation ?, we achieve that
If we note ?1=?-1 and ?2=?-1, we can see that the D-matrices provide a representation of the homogeneous Lorentz group:
There exist many of this kind of representations, just like the scalar D?=1, the vector D? ??=? ?? and lots of different type of tensor and spinor representations. These particular representations are irreducible, in the sense that it is not possible to by a choice of basis to reduce all D(?) to the same block-diagonal form, with two or more blocks. However, we do not require at this point that D(?) be irreducible; in general it is a block-diagonal matrix with an arbitrary array of irreducible representations in the blocks. That is, the index l here includes a label that runs over the types of particle described and the irreducible representations in the different blocks, as well as another that runs over the components of the individual irreducible representations.
After we saw how to construct fields satisfying the Lorentz transformation rules in eq. (4.1.6.) and (4.1.7.), we now can construct the interaction density as
Hx will be a scalar, using the eq. (4.1.2.), in case of when the constant coefficients gl1’…lN’,l1…lM are chosen in the way to be Lorentz covariant, so for all ? we will have
Here we will not use derivatives, because we note the derivatives of components of these fields as just additional sorts of field components. It is not much more difficult to find coefficients gl1’…lN’,l1…lM that satisfy the equation above, than that of using of Clebsch-Gordan coefficients to connect together different kind of representations of the three-dimensional rotation group to form rotational scalars.
The fields to satisfy the Lorentz transformation it is sufficient that
?ul?x+b;p?,?D??jnW?,p=p0/?p0 × lDll?expi(?p)?b ul x;p,?,n
??l?x+b;p?,?D??jnW?,p=p0/?p0 × lDll?exp-i(?p)?b ?l x;p,?,n
These two equations are the basic requirements to help us to calculate the ul and ?l coefficient functions in terms of finite numbers of free parameters.
Taking one thing with another three different types of proper orthochronous Lorentz transformation, we will use the two equations above in three steps:
For the first step we must take eq. (4.1.12.) and eq. (4.1.13.) with ?=1 and b optional. Now we observe that ul x;p,?,n and ?l x;p,?,n will take the following form
and so the fields will be Fourier transforms
? l+x=?,n2?-3/2d3 pulp,?,neip?x ap,?,n(4.1.16.)
? l-x=?,n2?-3/2d3 p?lp,?,ne-ip?x a?p,?,n(4.1.17.)
For arbitrary homogeneous Lorentz transformations ?, together with eq. (4.1.14.) and (4.1.15.) the equations (4.1.16.) and (4.1.17.) are satisfied if and only if
Next step is to take p=0 in the last two equations, and ? we will nominate a standard boost L(q) that takes a particle of mass m from rest to some fourmomentum q?, and so Lp=1 and
Therefore in this case, eq. (4.1.18.) and (4.1.19.) will give
ulq,?,n=m/q01/2lDllL(q)ul0,?,n(4.1.21.)?lq,?,n=m/q01/2lDllL(q)?l0,?,n(4.1.22.)We can know the functions ulp,?,n and ?lp,?,n for all p, when we know the number of amount ul0,?,n and ?l0,?,n for zero momentum, and for a given representation D(?) of the homogeneous Lorentz group.
Last step is when we take p=0, but in this case ? will be a Lorentz transformation with p?=0 and take ? as a rotation R, hence it follows that W?,p=R, and so equations (4.1.18.) and (4.1.19.) we can write
?ul0,?,nD??jnR=lDll(R)ul0,?,n(4.1.23.)and??l0,?,nD??jn*R=lDll(R)?l0,?,n(4.1.24.)which is equal to
?ul0,?,nJ??jn=lJllul0,?,n(4.1.25.)and??l0,?,nJ??jn*=-lJll?l0,?,n(4.1.26.)where J(j) and J are the angular-momentum matrices in the representations Dj(R) and D(R).
4.2. Causal Scalar Fields
4.3. Causal Vector Fields
4.4. Representation of the Lorentz group using SU(2)×SU(2)