_{1}

Segal’s chronometric theory is based on a space-time D, which might be viewed as a Lie group with a causal structure defined by an invariant Lorentzian form on the Lie algebra u(2). Similarly, the space-time F is realized as the Lie group with a causal structure defined by an invariant Lorentzian form on u(1,1). Two Lie groups G, GF are introduced as representations of SU(2,2): they are related via conjugation by a certain matrix Win Gl(4). The linear-fractional action of G on D is well-known to be global, conformal, and it plays a crucial role in the analysis on space-time bundles carried out by Paneitz and Segal in the 1980’s. This analysis was based on the parallelizing group U(2). In the paper, singularities’ general (“geometric”) description of the linear-fractional conformal GF-action on F is given and specific examples are presented. The results call for the analysis of space-time bundles based on U(1,1) as the parallelizing group. Certain key stages of such an analysis are suggested.

The Lie groups U(2) and U(1,1) are the two main objects to be dealt with in this paper. Introduce U(2) as the totality of all two by two matrices Z (complex entries allowed) which satisfy

Here 1 is the unit matrix.

Similarly, U(1,1) is the totality of all two by two matrices U which satisfy

Here s is the diagonal matrix with entries 1, −1.

Often, these two Lie groups (especially when they carry bi-invariant metric of Lorentzian signature―see [

Recall that the notion of a parallelization (of a space-time bundle―see [

According to quantum mechanics, each object is assigned its state (or wave function but this latter notion we better reserve for a more specialized situation, namely, after a parallelization has been applied). An elementary particle (it “lives” in a certain world W of events) is described by the set of its possible states. The latter set is a certain subspace of the section space (sections can later be specified as smooth, or square-integrable, etc.―this is not the main concern here) of a certain vector bundle over W. At this point, states are not, yet, number-valued (for a scalar particle) or C^{k}-valued (k > 1, for particles of non-zero spin). One way or the other, we then need to convert to parallelized sections (to wave functions, in other words).

The respective Hilbert space can then be determined. It has become an acknowledged way of modern theoretical physics to describe elementary particles and their interactions in terms of induced representations of the (respective) symmetry group. As it is put in [

Conventional quantum mechanics uses representations of the Poincare group, which are induced from its Lorentz subgroup as in Wigner’s seminal work, [

In general, the parallelization procedure is essentially defined by choice of the parallelizing (four-dimensional but not necessarily commutative) subgroup N of the group G. Here G is the symmetry group of the space-time W (in our studies, G is the (conformal) group SU(2,2), see below). Typically N is a finite cover of the original space- time W. In Segal’s (with co-authors) publications the mostly used parallelizations were the M-, and the D-ones. Onp.170 of the monograph [

In [

The Lie groups G, G_{F} are introduced as two equivalent representations of SU(2,2). Namely, G is composed of those 4 by 4 matrices g (with unit determinant), which satisfy

where

a diagonal matrix.

Introduce the 4 by 4 matrix W,

which is formed by the 2 by 2 blocks

It is clear that

Under the conjugation of the matrix S by W we get

which determines another copy (denote it by G_{F}) of SU(2,2). Namely, G_{F} is composed of those 4 by 4 matrices

The correspondence

is an isomorphism between Lie groups G, G_{F}.

Each element g of G can be viewed as a 4 by 4 matrix determined via 2 by 2 blocks A, B, C, D:

Similarly, each element _{F} is composed of the 2 by 2 matrices

The linear-fractional action

of G is known ([Se-1976, p.35]) to be defined on the entire D = U(2). The linear- fractional (locally-defined) action

of G_{F} on F = U(1,1) has been introduced in [

Given any two by two matrix M, let W(M) stand for

which is defined for every U in F. The mapping W is conformal but it is not used in this paper. Formula (1.9) is a special case (see [

is defined if and only if Z is outside of the torus T where T consists of all matrices K in D = U(2) of the form

with p, q being arbitrary complex numbers of length one.

The following fundamental statement has been proven in [

Theorem 1 (D-F commutative diagram). If

Remark 1. In [

One of the main goals of the current article is to prove the following

Theorem 2. Let _{F} and let U be in F. The image

Having in mind certain earlier findings (see [_{F} (with the Lie algebra R + su(1,1) + su(1,1)) of the world F. When arranging for the basis in the space of the scalar representation, instead of the “left” and the “right” Lie algebras su(2) (see [

The above indicated problematic is of great interest both for mathematics (covariance of wave equations, invariant forms in spaces of induced representations, classes of special functions, etc.) as well as for physics. Namely, in [

Notice that the matrix g(Z) is on the torus T if and only if

where a matrix K is of the form (1.11).

For any of the 2 by 2 matrices involved, denote their corresponding entries as follows:

Then the above (2.1) reads as the equality

of these two matrices with entries

In accordance with (2.2), the entries of L in (2.3) are L_{1}, L_{2}, L_{3}, L_{4}; they are expressed in terms of the entries of matrices A, Z, B in accordance with (2.2) and with the left side of (2.1). The entries of N in (2.3) are N_{1}, N_{2}, N_{3}, N_{4}; they are expressed in terms of the entries of matrices K, C, Z, and D in accordance with the right side of (2.1).

In what follows, it is assumed that (1.5) and (1.9) from Section 1 hold. To adequately understand the ongoing notation, the reader is referred to (1.8) from above. Let us start with

Lemma.

Proof. Let

where

The singularity of V (expressed as proportionality of its rows) is equivalent to the existence of a certain (not necessarily real) number q to satisfy the equality of the second rows in (2.3):

In other words, the matrix g(Z) is as follows:

Since the matrix (2.5) has to be an element of U(2), v = 0 holds. It means that

This last matrix has entries

Similarly, the singularity of the matrix (2.6) is equivalent to the existence of a certain (not necessarily real) number p to satisfy the equality of the first rows in (2.3). Again, g(Z) has to be on the torus T, and the entire (2.3) has to hold. Equality of the second rows in (2.3) forces the matrix

Let us now proceed with

Theorem 2. Let _{F} and let U be in F. The image

Proof. Let

This implies

If

Remark 2. Several examples of transformations

For a given

If a matrix K is the image of Z under g, then the equality

holds. Clearly, (3.1) is equivalent to

Additionally, let a matrix K be of the form (1.11). Since g is a bijection of D, the matrix

For a matrix U in

Theorem 3. (Description of singularities of

Let us continue to discuss (including examples―see below) the set (3.4) properties (in other words, to discuss a domain of a transformation

Corollary 1.

As the first example, consider the following one-parameter subgroup in G: each g is determined by blocks

Here c = ch(t/2), s = sh(t/2)―hyperbolic cosine and sine of a real parameter t. Assume that t is not zero (that is, g is not an identical map). Interestingly, the matrix

Proposition 1. Each matrix Z in (3.3) is of the form

The proof reduces to a (3.5)-based direct computation. Notice that for any (admissible in these circumstances) choice of parameters p, q, t, the expression (

Hence, the following statement holds.

Corollary 2. The set of all singular points of a transformation (3.5) is W-diffeomor- phic to a set which is a (two-dimensional) torus with acircle cut off it: this circle is determined by the equation pq = 1 in (3.6).

Recall [

Corollary 3.

Proof. Applying Theorem 3, compute W(Z), where Z is an element of the set Y. One gets W(Z) as the product RS where

As the second example, consider a two-parameter group А in G which is an (Abelian) subgroupА from the Iwasawa decomposition G = КAN. An arbitrary element min А is of the form

where the blocks (1.6) of the matrix g are as follows:

Here c = cht_{1}, s = sht_{1}, hyperbolic cosine and sine of a real parameter t_{1}. The blocks (1.6) of the matrix f are as follows:

Here c = cht_{2}, s = sht_{2}, hyperbolic cosine and sine of a real parameter t_{2}.

The following statement can be proven by a direct computation:

Proposition 2. For an arbitrary element m of the form (3.8), the matrix Z in (3.3) belongs to the torus T, given by (1.11). Namely,

where c_{1} = cht_{1}, s_{1} = sht_{1}, с_{2} = cht_{2}, s_{2} = sht_{2}. Notice that none of the denominators in (3.11) vanishes since

In other words, restriction onto T of a transformation (3.8) is a bijection of T. The mapping W is inapplicable to matrices (3.11).That is why (according to Theorem 3) the set

Corollary 4. Each transformation

Remark 3. Corollary 4 can be proven on the basis of Theorem 2: in this case each transformation m of the form (3.8) is a bijection of the set D\T onto itself. In other words, none of the matrices

Remark 4. Corollary 4 is coherent to the matrix

The action (1.8) of G_{F} on F = U(1,1) has been introduced in [_{F} on F has been determined. However, in [

Levichev, A. (2016) A Contribution to the DLF-Theory: On Sin- gularities of the SU(2,2)-Action in U(1,1). Journal of Modern Physics, 7, 1963-1971. http://dx.doi.org/10.4236/jmp.2016.715174