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23
УДК 167/168
J. Y. Feng, A. V. Levichev
More on the mathematics of the 3-Fold Model: embedding of the oscillator world L into Segal's compact cosmos D1
The DLF theory can be understood as an attempt to modify the Standard Model by flexing the Poincare symmetry to certain 7-dimensional symmetries. The D part of the theory is known as Segal's Chronometry which is based on compact cosmos D=U(2) with the SU(2,2) fractional linear action on it. The oscillator group is viewed as a subgroup LG of the conformal group G=SU(2,2) and certain LG-orbits L in D are studied. We prove existence of such L and of such an embedding of F=U(1,1) into D,that D differs from F by a certain torus whereas D differs from L by a circle on that torus. In the general U(p,q) vsU(p+q) case,the Sviderskiy formula is described - as a tribute to the late Oleg S. Sviderskiy (July 31 1969 - March 30 2011).
Keywords: Levichev's DLF-theory, Segal's Chronometric theory, conformal group SU(2,2)
Дж. Фенг, А. Левичев
Еще о математике Триединой Модели: вложение осцилляторного мира L в компактный космос Сигала D
DLF-теория может быть интерпретирована как попытка такой модификации Стандартной Модели, при которой Пуанкаре-симметрия ослабляется до определенной 7-мерной симметрии. D-составляющая этой теории является Хронометрией Сигала, основанной на компактном космосе D=U(2) с дробно-линейным действием SU(2,2) на нем. Осцилляторная группа LG является подгруппой конформной группы SU(2,2). Изучены некоторые ее орбиты L в D. Мы доказываем существование такой L и такого вложения группы F=U(1,1) в D, что D отличается от F определенным тором, а D отличается от L лишь окружностью на этом торе. В общей ситуации, когда даны U(p,q) и U(p+q), охарактеризована формула Свидерского - в честь Олега Сергеевича Свидерского (31 июля 1969 - 30 марта 2011).
Ключевые слова: DLF-теория Левичева, Хронометрическая теория Сигала, конформная группа SU(2,2)
1 Introduction
In the context of Segal's chronometric theory it has been shown2 that exact invariance under a 7-dimensional isometry group K of the Einstein static universe (the latter being the universal cover of the chronometric compact cosmos D) guarantees approximate Poincare invariance and that it is far beyond the accuracy of currently available devices to experimentally decide between the two types of invariance. Having this in mind, the DLF theory (see3) can be viewed as an attempt to modify the Standard Model by flexing the Poincare symmetry to three types of 7-dimensional symmetries (one of them being the K-symmetry).
In terms of the DLF terminology, the current article is primarily dedicated to the L-ingredient of the theory. The oscillator group will be introduced below as a subgroup Lgof the conformal group G=SU (2,2),whereas by L we will denote a certain Lc-orbit in D. The main inquiries and statements of our article are motivated by the (still pending) necessity to deal with L-based parallelizations (additionally to parallelizations of vector bundles introduced in1: see the relevant discussion in2). In the DLF context, the oscillator Lie algebra l has been introduced in Section 3. 3 of3. Our Section 2 introduces both l and (some of the) corresponding Lie groups.
Part of the content of Section 3 is a tribute to the late Oleg S. Sviderskiy (July 31 1969 - March 30 2011).
Overall, the current article provides mathematical justification to some of the claims of4.
2 Realizations of the oscillator group as subgroups in U(2,1)and SU(2,2)
Following5, we now introduce the oscillator Lie algebra as the totality of all matrices
lX-± Z —lX-±
Z IX4 Z lX-± Z —lX-±
(2. 1)
where z = X2 + ix3. Here (and below) variables xi,X2,X3,X4are real.
Introduce Lie algebra u(2,1) as the totality of all 3 by 3 matrices m (complex entries allowed) which satisfy
ms + sm* = 0, (2. 2)
where s a diagonal matrix with entries 1, 1,-1.
Proposition 1. Each matrix (2. 1) belongs to u(2,1). The straightforward proof is omitted.
Remark 1. Each matrix (2. 1) can be viewed as a linear combination of e1,e2,e3,e4. Namely,
i 0 — 1 0 1 0 0 i 0 0 0 0"
е1 = 0 0 0 , е2 = -1 0 1 , ез = i 0 —i , e4 = 0 i 0
л 0 —i. . 0 1 0. .0 i 0 . .0 0 0.
The commutation relations are as follows:
[e2,e4] = e3,[e4,e3] = e2, [e2,e3] = 2ea. The last relation differs by factor 2 from the one inSection 3. 3 of6. The two Lie algebras are isomorphic. The following notation is used below: z=x2+ix3, e=elx4, m = zz/2.
Theorem 1. The totality of all matrices
1 — m + ix1 z m — ixt
U= —ez e ez
—m + ix1 z 1 + m — ixt
(2. 3)
is a (closed) oscillator subgroup in U(2,1).
Proof (outline). The matrix (2. 3) is the product of exp (x4e4) and expn (in that order) where n is the linear combination of e1,e2,e3from above. This observation guarantees, already, that the totality of all matrices (2. 3) is a group. It is helpful however to determine precise expressions for the (2. 3) parameters of the product UU. They are not given here since we will only need these expressions in later study. Proposition 1 guarantees that it is a subgroup of U(2,1).
Remark 2. It is clear from (2. 3) that our oscillator subgroup has topology of S' times R3. The (standard) group operation for its universal covering group (with the R4 topology) can be found on p. 411 of7.
We now introduce oscillator Lie group as a subgroup Lgin G=SU(2,2). The latter group and its fractional linear action on U(2)
g(Z) = (AZ + B)(CZ + D)-1
(where an element g is determined by 2 x 2 blocks A, B, C, D) is defined in [Le-11a Section 6]. Consider the set of all 4 by 4 matrices g of the form
g = U+e-1 (2. 4)
(direct sum of a matrix (2. 3) with the 1 by 1 matrix e"1 where e = e1X4).
Theorem 2. The totality Lg of all matrices (2. 4) is a closed oscillator subgroup in G=SU(2,2). The LG-orbit L of the matrix -1 under fractional linear action is all D=U(2) but a certain circle (described below). The stationary LG-subgroup of -1 is the following set (of two elements): z=0,x=0, exp(2ix4)='.
Proof. The determinant of the (2. 3) matrix U is e which is why the matrix g (2. 4) is in SU(2,2). From (2. 4), we form the following matrices:
1 —т + ixt z т — ¿Xj 0 —m + ixt z 1 + rn — ixj 0
A = , B= , C= , D=
—ez е. ez 0 0 0. 0 e"1.
Fractional linear action applied to an element Z from U(2) (we then take Z = -1) results in:
(AZ + B)(C Z + D)"
= (
1 — m + ix1
—ez
= (■
Z +
m—ix1 0
ez
0
—m + ix1 z
0
0
Z +
1 +m—ix1 0
1+2m-2ix1'
—1 + 2m — 2ix1 —z
2 ez —e
1 ez
0 e(l + 2m — 2ix1)
Раздел 4. Естествознание и идеи наследия семьи Рерихов 2т — 1 — Нхг —2 ez
= и - 9 •
2 ez е^(2т — 1 + 2ix1)
In other words, an element M of the orbit is
2m — 1 — 2ix1 2 ez
—2 ez e2(2m — 1 + 2ixi)
, (2. 5)
where u = (м
4i+2m-2ixi
It is well-known that an arbitrary Zfrom U(2) can be viewed as
1 0 0 d
u4 + iu3 u2 + iu1
lUi ~ Un Ua ~ lUo
where |d| = l,and u12 + u22 + u32 + u42 — 1,
with real variables ui, U2, U3, U4.
Then (viewing Zas given) the equality Z = M (of two matrices) reads:
1 0 0 d
u4 + iu3 u2 + 1Щ
UU-± ~ U2 W4 — IW3
2m — 1 — 21^! —2 ez
2 ez e2(2m — 1 + 2ix1)
(2. 6)
Start with the equality of the first entries in the first rows:
2m-l-2ix1 _ 4m2+4xl~l~4ix1
U. + iu3 = ■
1 +2m-2ix1
(2m+l)2+4x2
The case U4 = 1 (which implies ui= U2= U3 = 0) has no solution, we will return to it later. Otherwise, the above is equivalent
4m*+4xi-l .
tou4 =-rr—? andu-г =
(2m+i)2+4x(
(2m+i)2+4x(
This system is solved by
-"3
г and 2m =
1-Ц32-И42 (1-u4)2+U|-
We are done with the first entries of the first rows. Now we proceed with the second
entries there:
, . -2ez . .
u7 + 1Щ =-(2. 7)
z 1 1+2Ж-21Ж!
We write x2,x3 in polar coordinates as follows:
и
х2 = г sine,х3 — г cos е. Recalle = cosx4 + i sinx4. Let N — (2m + l)2 + 4xf.
Then (2. 7) is equivalent to the system:
Nu-l — —2r[(2m + 1) cos(x4 — h) — 2x1 sin(x4 — h)], Nu2 = 2r[(2m + 1) sin(x4 — h) + 2 x1cos(x4 — h)].
Since r, m, X1 are already determined (in terms of the element Z), the values of both cos (X4— h),
sin(x4- h) are uniquely determined.
The equality of the first entries in second rows of (2. 6) is equivalent to:
sin(x4 +h) - i cos (x4+h) = d(iui - U2) / 2ru.
Since the right hand side is a complex number of length one, the left hand side is uniquely determined (in terms of the matrix Z).
We have thus determined all four parameters of (2. 5) in terms of the matrix Z. The equality of the second entries in second rows of (2. 6) is satisfied, too.
Overall, we have shown that all elements of U(2), which cannot be represented by (2. 5), form the circle
T1 01
0 d
. (2. 8)
This is the case where U4 = ' (which implies U'= U2= U3 = 0, and subsequently x_' = m = 0), and (2. 6) does not hold.
Regarding the stationary subgroup: based on (2. 5) simple calculation finishes the proof of Theorem 2.
Remark 3. Since this stationary subgroup is not an invariant one in Lg, the oscillator group (2. 4) does not induce a group operation on its homogeneous space L. Compare it with a different outcome in the U(',') case (see Remark 2 of Section 3).
The content of the next section is reproduced from8.
3 Embedding of U(1,1) into U(2) and generalizations to higher dimensions: the Sviderskiy formula
Let us start with a brief discussion of the general case. Formula n = sm (3. 1a)
sets up a linear bijection between vector spaces of Lie algebras u (p, q) and u (p + q): (3. 1a) is mentioned on p. 219 of9. Here s is a diagonal matrix with p ones and q negative ones on the principal diagonal and u (p, q) is the set of all p + q by p + q matrices m which satisfy our (2. 2) above, given nonnegative integers p and q. Obviously,
m = sn (3. 1b)
is the formula for the inverse mapping from u (p + q) onto u(p, q).
Formulas (3. 1a, b) might be viewed as giving canonical linear correspondence between u (p, q) and u (p + q) but how about correspondence between Lie groups U (p, q) and U (p + q)?
The research in this direction has been started (see10) by the second author together with late Oleg S. Sviderskiy (31 July 1969 - 30 March 2011). As a tribute to Oleg, it is now suggested that the formula for the canonical correspondence between groups U (p, q) and U (p + q) be known as the Sviderskiy formula; it is presented below as Theorem 3.
We first describe how U (1,1) sits in U(2). This is defined by the following function h from D=U(2): the image of a matrix
from U(2) is the matrix Vwith entries
Vl = d / Z4, V2 = Z2 / Z4, V3 = - Z3 / Z4,V4 = 1 / Z4;(3. 2)
Z1 Z 2
Z
Z 3 Z 4
here d is the determinant of Z. Notice that the determinant of V equalsz1 / Z4.
Proposition 2. The mapping (3. 2) is only undefined for elements Z on the torus Z1 =Z4= 0 in D=U(2). The image is the entire F=U(1,1). In terms of Lorentzian metrics (introduced in11 on both D and F) the mapping (3. 2) is conformal. The tangent mapping (or the differential of h) at the neutral element of D is exactly our (3. 1 b).
Here we only notice that correspondence (3. 2) is similar to the one established in12 whereas other details of the proof are to be presented elsewhere.
Remark 1. Significant part of what is discussed in this section, also makes sense in the SO (p, q) vs SO (p + q)context.
Remark 2. Having in mind the realization (2. 4) of the oscillator Lie group, the following seems to be one of the most natural ways to introduce Fg, a subgroup of G=SU(2,2) locally isomorphic to U(1,1). In the context of13, the blocks of a generic element in Fg are as follows: B=C=0, A=Z, D=q where Z is from U(1,1) and q2 times determinant of Zis one. If to proceed similarly to how we did in the proof of the above Theorem 2, then the homogeneous space F inherits the group operation from FG since the stability subgroup is central in FG. Namely, this stability subgroup consists of scalar matrices i, -i, 1, -1 and Fis isomorphic to U(1,1).
Proposition 3 below relates the three worlds together in a more specific way than it has been done in14and in15. In that Proposition 3 we view F=U(1,1) as a subset of D defined by (3. 2), whereas we choose another L rather than the one described in Theorem 2 and Remark 3 of Section 2.
Proposition 3. F < L < D, embeddings of manifolds.
Proof. Let us conjugate the oscillator subgroup (2. 4) in SU (2,2) in such a way that the resulting orbit of a certain matrix X has the following property: it contains all elements of U(2) with non-zero entry Z4. In other words, the analogue of the circle (2. 8) for this orbit will be contained in the torus Z= Z4= 0. It is easy to verify that the following element go in SU(2,2) takes the matrix
X =
" О -1
into negative one: blocks B and C of go both vanish, whereas
0 i i 0"
A = , D =
i 0. 0 i.
The inverse of go is its own negative, from where it follows that - goLGgo is an appropriate subgroup conjugate to the group Lg defined by (2. 4). The corresponding circle (points of which do not belong to the orbit) is the image of (2. 8) under go:
0 d 1 0
Clearly, this circle is on the torus (points of that to rus do not belong to F). Proposition 3 is proved.
We now proceed with the Sviderskiy formula which defines an embedding of U (p, q) into U (p + q) as manifolds. This mapping is defined as a fractional linear application of a certain 2n by 2n matrix W to (all) matrices in U (p, q); here n = p + q. The n by n blocks A, B, C, D of the matrix W are defined as follows:
A=D=
3=C=
I 0
p
0 0
0 0
0 I
q
where Ip(respectively, Iq) stand for the unit matrix of size p (respectively, of size q).
Theorem 3 (the Sviderskiy formUla). The fractional linear application of the above introduced matrix Wis defined for all matrices in U (p, q), and U (p, q) is in a one-to-one correspondence with its image. The inverse mapping is also defined as the fractional linear transformation (by the same matrix W).
The proof is to be presented elsewhere.
Remark 3. The above(3. 2) is a special case of the Sviderskiy formula.
Acknowledgments
As regards the first author, this project was funded [in part] by BU UROP.
Раздел 4. Естествознание и идеи наследия семьи Рерихов Примечания
1 Прим. ред.: Статья публикуется в авторской редакции А. В. Левичева на англ. яз.
2 Segal I. D., Vogan Z. Z. Chronometrie Particle Theory I: An Alternative to the Higgs Mechanism / with D. Vogan, Z. Zhou. 1995. Unpublished. URL: http: // dedekind. mit. edu (дата обращения: 27. 10. 2014).
3 Leviehev А. V. 2011 Pseudo-Hermitian realization of the Minkowski world through the DLF-theory PhysicaScripta N. 11-9
4 Paneitz S. M., Segal I. E. Analysis in space-time bundles I: General considerations and the scalar bundle // J. of functional analysis. 1982. Vol. 47. P. 78-142.
5 Levichev A. V. Pseudo-Hermitian realization of the Minkowski world through the DLF-theory // Physica Scripta. 2011. № 1. P. 1-97. Section 7. URL: http: // math. bu. edu (дата обращения: 17. 12. 2014).
6 Ibid.
7 Ibid.
8 Levichev А. V. Oscillator Lie algebra and algebras u(2), u(1,1), as a single matrix system in u(2,1). // Lie algebras, algebraic groups, and the theory of invariants: proceedings of the Summer School-Conference. Samara, Russia, June 8-15 2009. Samara, 2009. P. 32-34.
9 Levichev A. V. Pseudo-Hermitian realization of the Minkowski world through the DLF-theory.
10 Hilgert J., Hofmann K.-H., Lawson J. D. Convex Cones and Semigroups. Clarendon Press, 1989.
11 Левичев А. В. Новые возможности применения DLF-подхода в физике микромира. URL: http: // grani. agni-age. net (дата обращения: 29. 10. 2014).
12 Dubrovin B. A., Fomenko A. T., Novikov S. P. Modern Geometry - Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Springer, 1991.
13 Levichev A. V.,Sviderskiy O. S. Lie groups U(p,q) of matrices as a single system based on fractional linear transformations: I. General consideration and cases p+q = 2,3 // Contemporary Problems of Analysis and Geometry = Проблемы анализа и геометрии: Proceedings of the International Conference / Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences. Novosibirsk. P. 58-59.
14 Levichev A. V. Pseudo-Hermitian realization of the Minkowski world through the DLF-theory.
15 Ibid.
16 Ibid. Section 6.
17 Levichev A. V. Three symmetric worlds instead of the Minkowski space-time // Trans. RANS. Ser. MMM&C. 7. 2003. № 3-4. P. 87-93.
18 Levichev A. V. Pseudo-Hermitian realization of the Minkowski world through the DLF-theory. Theorem 6.
