2.1. Motivation
The Minkowski spacetime is a pseudo-unitary irreducible representation of the Lorentz symmetry. Its associated invariant is an indefinite vector norm of signature
. Each transformation acts on a four-vector as a (real)
matrix. It is this pseudo-unitary representation that reduces properly back to the reducible
dimensional representation of the Newtonian space and time. Such non-unitarity we see as the defining signature of spacetime physics. The full
invariant Fock space of the harmonic oscillator serves well as a solid picture of the single particle phase space under quantum mechanics, especially under the serious treatment of rigged Hilbert space formulation [
11,
12], giving full justice to the Hermitian nature of the position and momentum operators. A similar treatment of the
version could play an equivalent role in the proper formulation of Lorentz covariant quantum mechanics [
8]. One way or another, the essence of going from Newtonian physics to ‘relativistic’ physics should be like a direct consequence of embedding the Newtonian space and time into the Minkowski spacetime. It is then very desirable to have the
invariant Fock space, for the ‘three dimensional’ quantum harmonic oscillator problem, sits inside a full
invariant Fock space in a manner directly analogous to how the Newtonian space sits inside the Minkowski spacetime.
We want to have a complete set of Fock states with sensible norms as solutions to the problem, keeping the Lorentz symmetry while maintaining that there are four states transforming as a Minkowski four-vector. As an irreducible representation of the Lorentz group, the latter corresponds to a non-unitary one. However, it is the same non-unitarity of the Minkowski spacetime as a representation space. That is the natural framework to see the problem as a Lorentz covariant version of the rotational covariant picture of the ‘three dimensional’ theory. The indefinite Minkowski norm is what is preserved by the Lorentz transformations. We seek its natural extension in the form of pseudo-unitary norm for the Fock space, upon the restriction to the subspace of the four states.
2.2. Operator Representations and Fock States with Hermite Polynomials
We start at the level of symmetry or algebraic structure at the abstract level. The symbols
,
,
,
, …etc. are to be seen as abstract algebraic quantities, for which we seek a representation as operators on a Hilbert space. The relevant Lie algebra is that of
, given as
for which we focus on representations of the Lorentz symmetry with vanishing spins, i.e., its six generators
can be taken as
. The unitary representation as a direct extension of the
case, with only
and
, for standard quantum mechanics is straightforward [
13,
14]. Yet, at least when applied to the harmonic oscillator problem, the Fock state wavefunctions have undesirable behavior and divergence unless restricted to spacelike or timelike domains, under which there may be other mathematical issues for the full theory. Additionally, the integral inner products in either case contain a divergent volume factor that has to be artificially dropped for them to make sense. These have been well analyzed in Reference [
13], with their undesirable Lorentz transformation properties also well addressed in Reference [
7], as summarized above. The pseudo-unitary representation is obtained as
where, as operators on a space of functions of real variables
(
),
We have
, with
being the Kronecker delta symbol. Note that
and
, and hence
, are represented by anti-Hermitian operators, therefore we have a non-unitary representation of the group
or its subgroup
. For the complex combinations
and
, we have then
while
,
, satisfying
and
. The (total) number operator can be written as
and decomposed into a sum of the Hermitian number operators
, where
and
, easily seen from (
6). We have
and
. The normalized Fock states are eigenstates of
operators,
Solving (
8) in
coordinates, in which
we obtain the eigenstate wavefunctions as
where
,
are the standard Hermite polynomials. Hence, we have an explicit solution for a complete set of the Fock states wavefunctions without any problem of the other formulations.
In terms of
and
, the above is just like a quantum version of the harmonic oscillator in the four Euclidean classical dimensions. The Hilbert space spanned by the eigenstate wavefunctions looks completely conventional with an inner product giving a positive definite norm for the eigenstates in a usual manner. However, we only have to introduce the notation
and
to see that
corresponds exactly to the naively expected Hamiltonian of the covariant harmonic oscillator in Equation (
1). It is interesting to note that identifying
simply as
(and
as
), and the same for
, works too though the Hermitian
and
then differs from the representations of
and
with an
i factor. The non-unitary nature of the representation and a sensible notion of a pseudo-unitary inner product on the Hilbert space can be seen by looking into the eigenstates and their transformation properties under the Lorentz symmetry, which we turn to next.
2.3. Transformation Properties under the Lorentz Boosts
The Lorentz-algebra generators
are represented by the operators
, where
form a usual, unitarily represented
subalgebra of spatial rotations, while
are the anti-Hermitian boost operators. To examine the nature of the obtained states under the Lorentz transformation, we act with the boost generator in the, arbitrarily chosen,
direction on the eigenstate (
10). Using the properties of Hermite polynomials we get
We look into
level, as those four states should correspond to the components of a four vector. From (
12) we can obtain
as a matrix
Exponentiating, we get
the corresponding finite boost by the real parameter
. Alternatively, we can see the transformation as a rotation in
plane by a purely imaginary angle
. We find the action of
on arbitrary function
as
In particular,
while
and
are invariant as
is obviously invariant under any Lorentz transformation.
Seen differently, we can introduce
, with
, to show that in the basis formed by four
states,
takes the usual form
preserving a Minkowski norm on the real span of the four
vectors. The states hence transform as components of a Minkowski four-vector. In fact, that real span can actually be seen as a model of the Minkowski spacetime with the
invariant subspace spanned by the single
state and the complementary subspace spanned by the three
states, modeling the Newtonian time and space, respectively.
The Minkowski norm, or the extension of it to the complex span of the
vectors, and further to the whole Hilbert space spanned by all the Fock states, is definitely not unitary. We seek exactly an inner product, or rather an invariant bilinear functional [
15], different from the standard
corresponding to the
-norm for the wavefunctions, one that is invariant under any Lorentz transformation.
2.4. The Pseudo-Unitary Inner Product or Invariant Bilinear Functional
Fock states wavefunctions, given in Equation (
10), are orthonormal according to
, as the inner product is usually defined on a unitary Hilbert space. Label
n here is to be understood as
, and similar for
m. Therefore, we have
. Since the Lorentz transformations, boosts in particular, are not represented by unitary operators, such an inner product cannot be preserved in general, as can easily be seen from the results above, e.g., we have
Instead, we define another inner product given through the Fock state basis as
and extend it to the full vector space assuming sesqulinearity. It gives an indefinite norm, which is the natural extension of the Minkowski norm on the subspace of the real span of the four
vectors, and is invariant under the Lorentz transformations. In particular, for the boost
we have
Moreover, a state vector that is proportional to the sum or difference of the two states here above have zero norm under the inner product. We have, in general, spacelike, timelike, and lightlike state vectors with positive, negative, and vanishing pseudo-unitary norms, respectively. All vectors have finite norm and are all normalizable to the norm values of 1, 0, and −1, though the notion of normalization is an empty one for the lightlike states, obviously. It is important to note that normalizations with respect to
and
are in general not the same. All the basis Fock states are, however, normalized with respect to both, and none of them is lightlike.
Splitting the pseudo-unitary inner product notation
, one should consider the ket
as simply another notation for
, while the bra
as a linear functional is in general different from
. We have explicitly
which defines all
implicitly. We have then for the inner product
where we have used the fact that the wavefunctions
, given explicitly in Equation (
10), are odd and even in
for odd and even
, respectively. This gives a nice integral representation of it in terms of the wavefunctions (In some sense, it may be more proper to write things in terms of an alternative formulation of the wavefunctions as
. The latter is however a lot more clumsy to work with. Moreover, having two wavefunction representations of the states here only causes potential confusion.). Note that on the Hilbert space for a non-unitary representation of a noncompact group, there may not exist an invariant inner product. Certainly not a positive definite one. The wavefunctions of the states may not be squared integrable either. The appropriate structure to look for is an invariant bilinear functional [
15]. Our
inner product is exactly a gadget of that kind.
There is a simple way to write the mathematical relation between the two inner products that gives also an easy way to see the Lorentz invariance of the pseudo-unitary one. It is given in terms of a parity operator
, which sends
to
, as
We can actually take this as the definition. The
factor in our definition of the inner product in term of the Fock state basis above is exactly the
eigenvalue of
. With it, we have nicely
A good way to appreciate the Lorentz structure of the Hilbert space spanned by the Fock states is the following. We first look at the parallel for the case of the ‘three dimensional’ quantum harmonic oscillator. The three
states transform under
as components of an Euclidean three-vector. The
state is invariant. The two constant
n-level subspaces are vector spaces for the three dimensional defining representation and the trivial representations of
. For the
level, it corresponds exactly to the symmetric part of the product of two
representations, i.e., transforming as the Euclidean symmetric two tensor and the invariant (
). The standard
wavefunctions clearly show that, for an explicit check. One goes on to the higher
n-tensors for the higher
n levels. As also similarly discussed in Reference [
7], actually for the general Minkowski case, at the
n level, the full set of Fock states is a symmetric tensor of
which reduces to irreducible representations of
corresponding to the rank of the traceless tensors in the decomposition. The rank numbers are
n,
, …, (0 or 1). The pattern is essentially the same for any ‘
dimensional’ harmonic oscillator with the Fock states at level
n obtained by the action of
n creation operators on the
state, with the
independent creation operators transforming as a
-vector. The structure is not sensitive to the actual background signature
the latter has. Such representations, of
or
, are unitary only for the Euclidean case. For ‘three dimensional’ case, the rank of each of those traceless (Cartesian) tensors is exactly the
j value. Back to our
Lorentzian case, the finite dimensions of those traceless irreducible tensors are given by the square of rank plus one, with the full result explicitly illustrated in the next section. The way the Fock states for the ‘three dimensional’ states sit inside our Fock states at each
n-level can also be easily traced from the perspective of the Cartesian tensors.
The nature of the
n-level states as components of the symmetric
n-tensors can also be directly seen by looking at the wavefunctions given in terms of products of the Hermite polynomials with the common invariant factor
, which is essentially the
state wavefunction. It is then easy to appreciate the right pseudo-unitary inner product as given by Equation (
20) or Equation (
24). The norm as an invariant should better be expressed as
so that the upper indices in the wavefunction
can be contracted with the lower indices in the otherwise conjugate function
. For an Euclidean case
, as for the unitary inner product. To get to the pseudo-unitary inner product which goes along with the Minkowski nature of the tensors, it is then obvious that we only need to turn the
variable appearing in
, which are the tensors with lower indices, into
or
. The extra
i factor involved in the exact state for the
level as the component of a Minkowski four-vector, as discussed right above and in relation to Equation (
18), does not matter, due to the sequlinearity of the inner product. The invariant factor
, of course, does not change, though it is to be interpreted as
and
, accordingly.