8.1. Problems with Physical Interpretation of IRs
Consider first the case when the quantity
is very large. Then, as follows from Equations (
20) and (
21), the action of the operators
on states localized on
or
can be approximately written as
, respectively. Therefore a question arises whether the standard Poincare energy
E can be defined as
. Indeed, with such a definition, states localized on
will have a positive energy while states localized on
will have a negative energy. Then a question arises whether this is compatible with the standard interpretation of IRs, according to which the following requirements should be satisfied:
Standard-Interpretation Requirements: Each element of the full representation space represents a possible physical state for the given elementary particle. The representation describing a system of N free elementary particles is the tensor product of the corresponding single-particle representations.
Recall that the operators of the tensor product are given by sums of the corresponding single-particle operators. For example, if
is the operator
for particle 1 and
is the operator
for particle 2 then the operator
for the free system
is given by
. Here it is assumed that the action of the operator
(
) in the two-particle space is defined as follows. It acts according to Equation (
20) or (
21) over its respective variables while over the variables of the other particle it acts as the identity operator.
One could try to satisfy the standard interpretation as follows.
A) Assume that in Poincare approximation the standard energy should be defined as where the plus sign should be taken for the states with the carrier in and as the minus sign—for the states with the carrier in . Then the energy will always be positive definite.
B) One might say that the choice of the energy sign is only a matter of convention. Indeed, to measure the energy of a particle with the mass m one has to measure its momentum and then the energy can be defined not only as but also as . In that case the standard energy in the Poincare approximation could be defined as regardless of whether the carrier of the given state is in or .
It is easy to see that either of the above possibilities is incompatible with Standard-Interpretation Requirements. Consider, for example, a system of two free particles in the case when is very large. Then with a high accuracy the operators and can be chosen diagonal simultaneously.
Let us first assume that the energy should be treated according to B). Then a system of two free particles with the equal masses can have the same quantum numbers as the vacuum (for example, if the first particle has the energy and momentum while the second one has the energy and the momentum ) what obviously contradicts experiment. For this and other reasons it is well known that in Poincare invariant theory the particles should have the same energy sign. Analogously, if the single-particle energy is treated according to A) then the result for the two-body energy of a particle-antiparticle system will contradict experiment.
We conclude that IRs of the dS algebra cannot be interpreted in the standard way since such an interpretation is physically meaningless even in Poincare approximation. The above discussion indicates that the problem we have is similar to that with the interpretation of the fact that the Dirac equation has solutions with both, positive and negative energies.
As already noted, in Poincare and AdS theories there exist positive energy IRs implemented on the upper hyperboloid and negative energy IRs implemented on the lower hyperboloid. In the latter case Standard-Interpretation Requirements are not satisfied for the reasons discussed above. However, we cannot declare such IRs unphysical and throw them away. In QFT quantum fields necessarily contain both types of IRs such that positive energy IRs are associated with particles while negative energy IRs are associated with antiparticles. Then the energy of antiparticles can be made positive after proper second quantization. In view of this observation, we will investigate whether IRs of the dS algebra can be interpreted in such a way that one IR describes a particle and its antiparticle simultaneously such that states localized on are associated with a particle while states localized on are associated with its antiparticle.
By using Equation (
10), one can directly verify that the operators (
20) and (
21) are Hermitian if the scalar product in the space of IR is defined as follows. Since the functions
and
in Equation (
10) have the range in the space of IR of the su(2) algebra with the spin
s, we can replace them by the sets of functions
and
, respectively, where
. Moreover, we can combine these functions into one function
where the variable
can take only two values, say +1 or -1, for the components having the carrier in
or
, respectively. If now
and
are two elements of our Hilbert space, their scalar product is defined as
where the subscript
applied to scalar functions means the usual complex conjugation.
At the same time, we use
to denote the operator adjoint to a given one. Namely, if
A is the operator in our Hilbert space then
means the operator such that
for all such elements
and
that the left hand side of this expression is defined.
Even in the case of the operators (
20) and (
21) we can formally treat them as integral operators with some kernels. Namely, if
, we can treat this relation as
where in the general case the kernel
of the operator
A is a distribution.
As follows from Equations (
38–
40), if
then the relation between the kernels of these operators is
In particular, if the operator
A is Hermitian then
and if, in addition, its kernel is real then the kernel is symmetric,
i.e.,
In particular, this property is satisfied for the operators
and
in Equations (
20) and (
21). At the same time, the operators
which are present in Equations (
20) and (
21), are Hermitian but have imaginary kernels. Therefore, as follows from Equation (
42), their kernels are antisymmetric:
In standard approach to quantum theory, the operators of physical quantities act in the Fock space of the given system. Suppose that the system consists of free particles and their antiparticles. Strictly speaking, in our approach it is not clear yet what should be treated as a particle or antiparticle. The considered IRs of the dS algebra describe objects such that
is the full set of their quantum numbers. Therefore we can define the annihilation and creation operators
for these objects. If the operators satisfy the anticommutation relations then we require that
while in the case of commutation relations
In the first case, any two
a-operators or any two
operators anticommute with each other while in the second case they commute with each other.
The problem of second quantization can now be formulated such that IRs should be implemented as Fock spaces,
i.e., states and operators should be expressed in terms of the
operators. A possible implementation is as follows. We define the vacuum state
such that is has a unit norm and satisfies the requirement
The image of the state
in the Fock space is defined as
and the image of the operator with the kernel
in the Fock space is defined as
One can directly verify that this is an implementation of IR in the Fock space. In particular, the commutation relations in the Fock space will be preserved regardless of whether the
operators satisfy commutation or anticommutation relations and, if any two operators are adjoint in the implementation of IR described above, they will be adjoint in the Fock space as well. In other words, we have a
homomorphism of Lie algebras of operators acting in the space of IR and in the Fock space.
We now require that in Poincare approximation the energy should be positive definite. Recall that the operators (
20) and (
21) act on their respective subspaces or in other words, they are diagonal in the quantum number
.
Suppose that
and consider the quantized operator corresponding to the dS energy
in Equation (
20). In Poincare approximation,
is fully analogous to the standard free energy and therefore, as follows from Equation (
50), its quantized form is
This expression is fully analogous to the quantized Hamiltonian in standard theory and it is well known that the operator defined in such a way is positive definite.
Consider now the operator
. In Poincare approximation its quantized form is
and this operator is negative definite, what is unacceptable.
One might say that the operators and are “nonphysical”: is the operator of object’s annihilation with the negative energy, and is the operator of object’s creation with the negative energy.
We will interpret the operator as that related to antiparticles. In QFT the annihilation and creation operators for antiparticles are present in quantized fields with the coefficients describing negative energy solutions of the corresponding covariant equation. This is an implicit implementation of the idea that the creation or annihilation of an antiparticle can be treated, respectively as the annihilation or creation of the corresponding particle with the negative energy. In our case this idea can be implemented explicitly.
Instead of the operators
and
, we define new operators
and
. If
is treated as the “physical" operator of antiparticle annihilation then, according to the above idea, it should be proportional to
. Analogously, if
is the “physical" operator of antiparticle creation, it should be proportional to
. Therefore
where
is a phase factor such that
As follows from this relations, if a particle is characterized by additive quantum numbers (e.g., electric, baryon or lepton charges) then its antiparticle is characterized by the same quantum numbers but with the minus sign. The transformation described by Equations (
53) and (
54) can also be treated as a special case of the Bogolubov transformation discussed in a wide literature on many-body theory (see, e.g., Chapter 10 in Reference [
44] and references therein).
Since we treat
as the annihilation operator and
as the creation one, instead of Equation (
48) we should define a new vacuum state
such that
and the images of states localized in
should be defined as
In that case the
operators should be such that in the case of anticommutation relations
and in the case of commutation relations
We have to verify whether the new definition of the vacuum and one-particle states is a correct implementation of IR in the Fock space. A necessary condition is that the new operators should satisfy the commutation relations of the dS algebra. Since we replaced the
operators by the
operators only if
, it is obvious from Equation (
50) that the images of the operators (
20) in the Fock space satisfy Equation (
4). Therefore we have to verify that the images of the operators (
21) in the Fock space also satisfy Equation (
4).
Consider first the case when the operators
satisfy the anticommutation relations. By using Equation (
53) one can express the operators
in terms of the operators
. Then it follows from the condition (
53) that the operators
indeed satisfy Equation (
57). If the operator
is defined by Equation (
50) and is expressed only in terms of the
operators at
, then in terms of the
-operators it acts on states localized in
as
As follows from Equation (
57), this operator can be written as
where
C is the trace of the operator
:
In general,
C is some indefinite constant. It can be eliminated by requiring that all quantized operators should be written in the normal form or by using another prescriptions. The existence of infinities in the standard approach is the well known problem and we will not discuss it. Therefore we will always assume that if the operator
is defined by Equation (
50) then in the case of anticommutation relations its action on states localized in
can be written as in Equation (
60) with
. Then, taking into account the properties of the kernels discussed above, we conclude that in terms of the
-operators the kernels of the operators
change their sign while the kernels of the operators in Equation (
44) remain the same. In particular, the operator
acting on states localized on
has the same kernel as the operator
acting on states localized in
has in terms of the
a-operators. This implies that in Poincare approximation the energy of the states localized in
is positive definite, as well as the energy of the states localized in
.
Consider now how the spin operator changes when the
a-operators are replaced by the
b-operators. Since the spin operator is diagonal in the variable
, it follows from Equation (
60) that the transformed spin operator will have the same kernel if
where
is the kernel of the operator
. For the
z component of the spin operator this relation is obvious since
is diagonal in
and its kernel is
.If we choose
then the validity of Equation (
62) for
can be verified directly while in the general case it can be verified by using properties of
symbols.
The above results for the case of anticommutation relations can be summarized as follows. If we replace
by
in Equation (
21) then the new set of operators
obviously satisfies the commutation relations (
4). The kernels of these operators define quantized operators in terms of the
-operators in the same way as the kernels of the operators (
20) define quantized operators in terms of the
-operators. In particular, in Poincare approximation the energy operator acting on states localized in
can be defined as
and in this approximation it is positive definite.
At the same time, in the case of commutation relation the replacement of the
-operators by the
-operators is unacceptable for several reasons. First of all, if the operators
satisfy the commutation relations (
47), the operators defined by Equation (
53) will not satisfy Equation (
58). Also, the r.h.s. of Equation (
60) will now have the opposite sign. As a result, the transformed operator
will remain negative definite in Poincare approximation and the operators (
44) will change their sign. In particular, the angular momentum operators will no longer satisfy correct commutation relations.
We have shown that if the definitions (
48) and (
49) are replaced by (
55) and (
56), respectively, then the images of both sets of operators in Equation (
20) and Equation (
21) satisfy the correct commutation relations in the case of anticommutators. A question arises whether the new implementation in the Fock space is equivalent to the IR described in
Section 5. For understanding the essence of the problem, the following very simple pedagogical example might be useful.
Consider a representation of the SO(2) group in the space of functions on the circumference where is the polar angle and the points and are identified. The generator of counterclockwise rotations is while the generator of clockwise rotations is . The equator of the circumference contains two points, and and has measure zero. Therefore we can represent each as a superposition of functions with the carriers in the upper and lower semi circumferences, and . The operators A and B are defined only on differentiable functions. The Hilbert space H contains not only such functions but a set of differentiable functions is dense in H. If a function is differentiable and has the carrier in then and also have the carrier in and analogously for functions with the carrier in . However, we cannot define a representation of the SO(2) group such that its generator is A on functions with the carrier in and B on functions with the carrier in because a counterclockwise rotation on should be counterclockwise on and analogously for clockwise rotations. In other words, the actions of the generator on functions with the carriers in and cannot be independent.
In the case of finite dimensional representations, any IR of a Lie algebra by Hermitian operators can be always extended to an UIR of the corresponding Lie group. In that case the UIR has a property that any state is its cyclic vector
i.e., the whole representation space can be obtained by acting by representation operators on this vector and taking all possible linear combinations. For infinite dimensional IRs this is not always the case and there should exist conditions for IRs of Lie algebras by Hermitian operators to be extended to corresponding UIRs. This problem has been extensively discussed in the mathematical literature (see e.g., References [
29,
30,
31,
32]). By analogy with finite dimensional IRs, one might think that in the case of infinite dimensional IRs there should exist an analog of the cyclic vector. In
Section 7 we have shown that for infinite dimensional IRs of the dS algebra this idea can be explicitly implemented by choosing a cyclic vector and acting on this vector by operators of the enveloping algebra of the dS algebra. Therefore if IRs are implemented as described in
Section 5, one might think that the action of representation operators on states with the carrier in
should define its action on states with the carrier in
.
8.2. Example of Transformation Mixing Particles and Antiparticles
We treated states localized in
as particles and states localized in
as corresponding antiparticles. However, the space of IR contains not only such states. There is no rule prohibiting states with the carrier having a nonempty intersection with both,
and
. Suppose that there exists a unitary transformation belonging to the UIR of the dS group such that it transform a state with the carrier in
to a state with the carrier in
. If the Fock space is implemented according to Equations (
48) and (
49) then the transformed state will have the form
while with the implementation in terms of the
operators it should have the form (
56). Since the both states are obtained from the same state with the carrier in
, they should be the same. However, they cannot be the same. This is clear even from the fact that in Poincare approximation the former has a negative energy while the latter has a positive energy.
Our construction shows that the interpretation of states as particles and antiparticles is not always consistent. We can only guarantee that this interpretation is consistent when we consider only states localized either in
or in
and only transformations which do not mix such states. In quantum theory there is a superselection rule (SSR) prohibiting states which are superpositions of states with different electric, baryon or lepton charges. In general, if states
and
are such that there are no physical operators
A such that
then the SSR says that the state
is prohibited. The meaning of the SSR is now widely discussed (see e.g., Reference [
45] and references therein). Since the SSR implies that the superposition principle, which is a key principle of quantum theory, is not universal, several authors argue that the SSR should not be present in quantum theory. Other authors argue that the SSR is only a dynamical principle since, as a result of decoherence, the state
will quickly disappear and so it cannot be observable.
We now give an example of a transformation, which transform states localized in
to ones localized in
and
vice versa. Let
be a matrix which formally coincides with the metric tensor
. If this matrix is treated as a transformation of the dS space, it transforms the North pole
to the South pole
and
vice versa. As already explained, in our approach the dS space is not involved and in
Section 5,
Section 6 and
Section 7 the results for UIRs of the dS group have been used only for constructing IRs of the dS algebra. This means that the unitary operator
corresponding to
I is well defined and we can consider its action without relating
I to a transformation of the dS space.
If
is a representative defined by Equation (
17) then it is easy to verify that
and, as follows from Equation (
13), if
is localized in
then
will be localized in
. Therefore
transforms particles into antiparticles and
vice versa. In
Section 3 we argued that the notion of the spacetime background is unphysical and that unitary transformations generated by self-adjoint operators may not have a usual interpretation. The example with
gives a good illustration of this point. Indeed, if we work with the dS space, we might expect that all unitary transformations corresponding to the elements of the group SO(1,4) act in the space of IR only kinematically, in particular they transform particles to particles and antiparticles to antiparticles. However, in QFT in curved spacetime this is not the case. Nevertheless, this is not treated as an indication that standard notion of the dS space is not physical. Although fields are not observable, in QFT in curved spacetime they are treated as fundamental and single-particle interpretations of field equations are not tenable (moreover, some QFT theorists state that particles do not exist). For example, as shown in References [
9,
10,
11,
46,
47], solutions of fields equations are superpositions of states which usually are interpreted as a particle and its antiparticle, and in the dS space neither coefficient in the superposition can be zero. This result is compatible with the Mensky’s one [
27] described in the beginning of this section. One might say that our result is in agreement with those in dS QFT since UIRs of the dS group describe not a particle or antiparticle but an object such that a particle and its antiparticle are different states of this object (at least in Poincare approximation). However, as noted above, in dS QFT this is not treated as the fact that the dS space is unphysical.
The matrix I belongs to the component of unity of the group SO(1,4). For example, the transformation I can be obtained as a product of rotations by 180 degrees in planes and . Therefore, can be obtained as a result of continuous transformations when the values of and change from zero to . Any continuous transformation transforming a state with the carrier in to the state with the carrier in is such that the carrier should cross at some values of the transformation parameters. As noted in the preceding section, the set is characterized by the condition that the standard Poincare momentum is infinite and therefore, from the point of view of intuition based on Poincare invariant theory, one might think that no transformation when the carrier crosses is possible. However, as we have seen in the preceding section, in variables the condition defines the equator of corresponding to and this condition is not singular. So from the point of view of dS theory, nothing special happens when the carrier crosses . We observe only either particles or antiparticles but not their linear combinations because Poincare approximation works with a very high accuracy and it is very difficult to perform transformations mixing states localized in and .