5.1. Historical Analogy as a Guide to Generalisation
To construct a mathematical model of spin measurements in smeared-space, we proceed by analogy with the historical development of canonical QM. (See
Appendix C.1 for details.) Hence, we seek a set of constant-valued matrices
that satisfy the same algebraic structures as the components of angular momentum
.
In the canonical theory, the relevant algebra for the angular momentum operators is simply the three-dimensional rotation algebra,
, scaled by a factor of
ħ (A26). However, in the smeared-space model, the situation is more complicated. In
Section 4.3, we showed how the smeared-space angular momentum operators can be decomposed into the sum of four terms: a canonical quantum term
acting on the first subspace of the tensor product state
(
54), a “pure” geometric part
acting on the second, and two “cross terms”,
and
, acting on both subspaces (
74)–(
75). The subcomponents
were found to obey the subalgebra defined by Equations (
76)–(
83). Together, these equations ensure that the rescaled
Lie algebra, with
(
40), holds for
. In addition, we used the alternative definition
(
85), leading to the subalgebra (
86)–(
90) for
.
Hence, when searching for generalised spin operators, whose eigenvalues are to be interpreted as the possible spins of the composite matter-plus-geometry quantum state, we have three possible options to explore. First, we may search for exact analogue Equations (
74) and (
75). This requires
to be decomposed into the sum of four terms,
, where
,
,
and
. In this case,
and
are required to be finite-dimensional constant-valued matrices, acting on the first spin-subspace of the smeared tensor product state, that satisfy the
ħ-scaled Heisenberg algebra:
,
,
. (Here,
is used to denote the tensor product of the two spin subspaces, corresponding to matter and geometry, respectively.) Similarly,
and
must be finite-dimensional constant-valued matrices, acting on the second subspace of the tensor product, satisfying the
-scaled Heisenberg algebra:
,
,
. (The requirement that each representation of the Heisenberg algebra acts on a different subspace of the product state also ensures that
,
,
and
.)
However, it is straightforward to show that no such matrices exist. The matrices most similar to those we require are finite-dimensional representations of the Heisenberg group [
35]. This group has one central element (
z) and two sets of generators, usually denoted
and
by analogy with the canonical commutation relations, that satisfy the following algebra:
,
,
and
,
. In other words, while the central element
z commutes with all other generators, it is
not the identity element. Perhaps confusingly, the previous commutation relations are also typically referred to as the “Heisenberg algebra” in the mathematical literature, since they are the algebra of the Heisenberg group. However, they are
not equivalent to the position-momentum commutation relations of canonical QM [
35]. Therefore, this procedure fails, as it is impossible to define exact finite-dimensional analogues of the
subcomponents
.
Second, we may search for an alternative set of finite-dimensional constant-valued matrices,
, satisfying the relevant algebra. By the argument above, these cannot be defined in terms of finite-dimensional analogues of the position and momentum operators, i.e.,
,
, etc. Formally, we require
to satisfy an algebra analogous to (
76)–(
83) under the interchange
,
,
and
. In this case, we must again require that
act on the first subspace of the tensor product state, that
act on the second subspace, and that
and
act on both subspaces simultaneously. With this in mind, we note that the most natural operator that is able to act nontrivially on both spin subspaces is of the form
. However, it is straightforward to show that, using this definition,
and
, so that the analogues of Equations (
82) cannot be satisfied. Therefore, this procedure also fails.
Third, we may search for a smaller set of finite-dimensional constant-valued matrices,
, satisfying an analogue of the algebra (
86)–(
90) under the exchange
,
and
. Based on our previous considerations, this is clearly the most promising route. In the following section, we explore this possibility and construct explicit representations of the generator subcomponents
,
and
.
5.2. Generalised Algebra and GURs
Considering the arguments presented above, we define the generalised spin operator
as
where
and
are given by
and
Equations (
96) represent an extension of Equation (A37), which holds only for the matter sector. The prime on the Pauli operators acting on the second spin-subspace, corresponding to the spin part of the quantum state associated with the background geometry, indicates that this may posses a different fundamental spin to the matter component,
. In this case, the two spin subspaces have different dimensions. (In other words, we use the shorthand notations
,
and
,
.) It follows from the definitions (
94)–(
96) that
and
for any
s,
.
This is clearly the analogue of the operator (**) introduced below Equations (
86)–(
90). Using the identity
, Equations (
97)–(
99) are sufficient to show that the relations
also hold for any values of
s and
. Hence, in order to recover a rescaled spin Lie algebra for the generalised operators
(with
), we require the following commutation relations to hold between the cross terms
and
:
In this section, our main aim is to describe the generalised spin physics of electrons in smeared-space. Hence, since the situation in which
is of greatest physical interest, we restrict ourselves to this from now on. We then have
and
where
denotes the anti-commutator, which are equivalent to the canonical Equations (A39) and (A40), respectively. It is then straightforward to show that Equation (
102) holds if and only if
so that
However, unlike Equations (
97)–(
98) and (
100)–(
101), these relations hold only for
. (See
Appendix C.2.) Hence, consistency of the generalised spin structure implies that the quantum state associated with the background geometry must be
fermionic, with spin values
.
The generalised spin algebra for the subcomponents
is, therefore
Together, Equations (107)–(111) give rise to the rescaled
Lie algebra:
and the rescaled Clifford algebra:
for the generalised spin-measurement operators
(
94). From (112), it also follows that
Note that, in the limit
, the
term is not necessary to maintain the canonical Lie algebra structure. Since both
and
are representations of the
generators, and these representations commute with each other (
95), the combination
also satisfies the
algebra
if both sets of generators are weighted by the same scale factor. In this case, we may pull a single factor of
ħ outside the sum of terms on right-hand sides of the commutation relations, yielding
. However, in the presence of a two-scale theory, which is an essential feature of the smeared-space model [
22], the presence of
is unavoidable. Without it, it is not possible to construct an operator
that includes commuting representations of
weighted by different scale factors, i.e.,
and
(
), and which also satisfies a canonical-type commutation relation. In this case, it is not possible to pull a single factor (with units of action) outside the expression on the right-hand side of the relation
without including
(
99) in the definition of
(
94).
This is a fundamental difference between canonical two-particle states and the bipartite matter-plus-geometry states of the smeared-space model. Furthermore, it has clear physical interpretation. The first copy of the algebra, weighted by ħ, is generated by the spin of the matter sector, whereas the second copy, weighted by , is generated by the intrinsic spin of the background. If these spins are left to evolve freely, without interacting, the introduction of a second quantisation scale for geometry, , breaks the invariance of the composite matter-plus-geometry state. However, the spins do not evolve freely but interact via the cross term . The interaction is such that symmetry is restored, for the composite state, under a simple rescaling .
Written explicitly, the generalised spin matrices take the form:
and
is given by
This follows from the fact that the matrices are involutions. Hence, in the smeared-space model, are the analogues of the canonical spin- Pauli matrices, . However, unlike the canonical Pauli matrices, depend explicitly on both quantisation scales, ħ and .
It is straightforward to verify that all three spin operators
(115) have the eigenvalues:
which, for
, correspond to the following (un-normalised) eigenvectors:
The normalised eigenvectors of
may then be written as
and
where
Hence, the single electron plus smeared-background system has four spin states, as opposed to the two spin states of electrons on the fixed background of canonical QM. However, the operators and that act on the composite system have only two distinct sets of eigenvalues, . Each pair of eigenvalues has a 2-fold degeneracy, corresponding to one separable state and one state in which the spins of the electron and the background are entangled. The eigenvectors and correspond to spin “up” states, according to the measured values of and , whereas and correspond to spin “down” states.
For the unentangled states, , the spins of the matter and geometry components of the tensor-product smeared-state, and , are aligned. The spin up state is characterised by the individual values and the spin down state by the values . However, for the entangled eigenvectors, , there is no simple relation between the matter and geometry components of the total quantum state. Remarkably, the entangled eigenstates (119) have the same eigenvalues as the simple separable states (118).
We also note that, in the absence the interaction term
, the eigenvalues of the composite operator
are
. These correspond to the eigenvectors
, respectively, which in the limit
yield the familiar spin eigenvectors of a canonical two-particle state [
28]. Thus, the introduction of
not only restores
symmetry in the composite matter-plus-geometry system, in the presence of a two-scale theory with
, but also alters
two of the four spin-eigenstates while leaving the remaining two unchanged. This, in turn, shifts the corresponding eigenvalues by
just the right amount to introduce 2-fold degeneracy in the measured values of
and
.
A priori, there was no reason for us to anticipate that the additional terms required to restore
symmetry, i.e., those involving
in the algebra (
107)–(111), would simultaneously introduce degeneracy of the resulting spin states. However, had this
not been the case, the doubling of the spin degrees of freedom would, in principle, have been directly detectable via simultaneous measurements of
and
. This would have caused severe problems for the smeared-space model, at least philosophically, even if no mathematical inconsistencies were introduced. It is straightforward to see why.
In the non-spin part of the model, the doubling of the canonical degrees of freedom is detectable only indirectly, via the additional statistical fluctuations it induces in the measured values of position, momentum and angular momentum, etc. These generate the GURs discussed in previous sections. In effect, we assume a measurement scheme in which measurements can be made only on material bodies in space [
22]. Hence, we do not have direct physical access to the quantum degrees of freedom of the background, which can be detected only indirectly via their influence on quantum particles. Mathematically, this is expressed by tracing out, or, equivalently, integrating out the degrees of freedom in the first subspace of the tensor product Hilbert space, as in Equation (
13). However, since the spin part of the composite matter-plus-geometry state is finite-dimensional, no integrals appear anywhere in the corresponding formulae. Furthermore, there is no clear physical justification for tracing out half of the doubled spin degrees of freedom, since this would require us to make an arbitrary choice, i.e., which
two of the four possible spin states should we regard as physical?
Remarkably, the algebra (
107)–(111) saves us from this dilemma, just as it “saves” the
symmetry of the two-scale quantisation scheme. The resulting generalised spin model is both mathematically consistent
and consistent with the physical assumptions underlying the smeared-space model as a whole, despite the doubling of the number of dimensions in the spin Hilbert space.
Finally, we consider the GURs implied by the generalised spin algebra (
107)–(111). By analogy with Equation (
93),
takes the form:
Multiplying by the equivalent expression for
, we obtain the GUR for spin measurements in smeared-space. Again, it is beyond the scope of this paper to investigate the consequences of this relation in detail. Nonetheless, we note that it is of the general form:
where the leading contribution to the terms in the middle is of the form
. This is equivalent to the canonical uncertainty relation for spin measurements. The additional terms are non-canonical and depend on the ratio of the dark energy density to the Planck density, which determines the value of the geometry quantisation scale,
.
5.3. Generalised Gamma Matrices
The construction of a full theory of quantum dynamics in smeared Minkoswki space, i.e., quantum field theory on a smeared space-time background, lies well beyond the scope of the present work. Nonetheless, the results of previous sections allow us to make limited conjectures about the description of relativistic electrons in such a theory. In particular, our previous results suggest that the kinetic term in the usual Dirac Equation (A45) should be mapped according to
where
and
Less obvious is what happens to the canonical mass term, . In the momentum space of classical relativistic dynamics, the quantity analogous to the boost-invariant space-time interval, , is the invariant length of the 4-momentum vector. Up to factors of , this is simply the mass of the particle traversing the interval s: .
In canonical QFT, in which the space-time remains classical and sharply-defined, Lorentz invariance is preserved exactly. The classical mass appears as a parameter in the theory and is not promoted to the status of a quantum mechanical operator [
24]. However, in a consistent theory of smeared space-time, we expect a radically different scenario. Intuitively, we would expect an appropriate smearing procedure to introduce an irremovable minimum uncertainty in the length of a space-time interval,
. Consistency of the position and momentum space pictures should then imply a corresponding minimum uncertainty in the length of the 4-momentum vector,
. This is possible if the classical parameter
m is promoted to the status of a Hermitian operator,
.
In [
22], it was shown how to incorporate the effects of smearing directly into the definitions of observables. The resulting “smeared” Hermitian operators then act on the canonical quantum state
. This formulation of the model yields exactly the same predictions as the smeared-state picture in which the fundamental state is
. However, in the smeared-operator picture, classical isometries are mapped to superpositions of isometries in the extended phase space of theory [
22]. So far, this method has only been applied to the translation generators of classical Euclidean space, but it may, in principle, be extended to the generators of other symmetries. Hence, we will address ways to implement smeared Lorentz symmetry, using this method, in a future publication. We hope that such an approach may be capable of yielding a natural definition of the mass operator
.