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Proceedings of MEASURING ENERGY FIELDS: International Scientific ConferenceKamnik, Tunjice, 13-14 October 2007, 72-75 / edited by Igor Kononenko;- Kamnik : Zavod Zdravilni gaj, 2007ISBN 978-961-92200-0-9

DLF-approach as the development of Segal's chronometric theory. III: More on the tachyonic component
Alex Levichev,
Sobolev Institute of Mathematics, Novosibirsk, Russia; Boston University, Boston, USA e-mail: levit@math.nsc.ru and alevichev@gmail.com
Let G denote the conformal group SU(2,2). Recall the well-known linear-fractional G-action on D: g(z) = (Az + B)(Cz + D)-1 (1.1) where an element g is determined by 2â2 blocks A, B, C, D. This action is canonically lifted to the G~action on D~ (the latter action preserves the causal structure). Proofs of the above statements can be found in [Se-76, PaSe-82a]. Theorem 1 ([Al-76, Se-76]). If a bijection f of D~ preserves its causal structure then f is an element of the transformation group G~, determined by the action (1.1). In other words, the geometry of such a spacetime is determined by its causal structure ­ a fundamentally important property! Vector fields X0 = L-10, X1 = L14 ­ L23, X2 = L24 ­ L31, X3 = L34 ­ L12 form a left-invariant orthonormal basis on D = U(2), where fifteen vector fields Lij (index i is less than j, and they take on values -1, 0, 1, 2, 3, 4) are determined on D by the action (1.1). Let us keep the same symbols to denote respective vector fields on D~. Globally, D~ is R1 â S3, where S3 is represented by the group SU(2). A cosmological model based on D~ has been considered by many experts "to be an excellent model for the large-scale gravitational structure of the universe. It fell into disfavor only because of the belief, explicitly stated by Hubble, that there would be no redshift in it" ([Se-85, p.214]). Segal's chronometric theory explains redshift by the excess of D-energy over Menergy. That is why "there is no reason not to use the Einstein Universe D~ as a gravitational model, as originally proposed". Also, see [DS-01]. In Chronometry, there is a conformal invariant R, interpreted as the radius of a three-dimensional (spherical) space. Irving Segal has put it for the (long wanted by Dirac and others) third fundamental constant ­ additionally to the speed of light and to the Planck's constant. If (for mathematical convenience) one takes R=1, then the scalar curvature is 6. The conformally covariant wave operator is X02 ­ X12 ­ X22 ­ X32 + 1,

Abstract Segal's theory is outlined briefly. It is based on space-time D. There are space-times L and F, which are on equal footing with D. The main idea of the DLF-approach is that each object of nature has not just D-properties (that is, "conventional" ones) but L- and F-properties, too. The F-part of the theory is studied in more detail. 1. Introduction and the world D The importance of the Biological Field which is "a combination of different types of fields...of known and unknown origin..." has been stressed in [K-02] (p.237 and elsewhere). To try to understand those "unknown origins" we build a certain theoretical model, first. Otherwise, as someone said, "We do not have any idea how to detect those new fields, do not know what to measure." The current article is dedicated to the DLF-approach which is based on Segal's chronometric theory (see more details in [Le-03] and/or in [KL-05]). Having in mind the topic of the Conference, it is worth mentioning that significant part of paragraph I.3 of [KL-05] is dedicated to rigorous mathematical notion of energy. The DLF-theory forces to consider three types of energy instead of just one. The chronometric theory (see surveys [Le-93], [Le-95]) has been presented in dozens of articles many of which have been published in leading mathematical, physics, and astronomy journals. Here are some building blocks of the DLFapproach. Denote by M the Minkowski space-time (in its Hermitian realization, see Section 3 below, where the Caley map formula is given). Let D stand for the unitary group U(2). The image c(M) of the Caley map (refer to [Se-76] or to [Le-95]) is a dense open subset in D. Let us view M as a vector group. It is commutative: each left translation is the respective right translation, too. The family {Cy} of subsets in M forms a bi-invariant cone field; each Cy = y + C, where C is a light cone at the origin of M. Due to the presence of the Caley map, there is the corresponding cone field on D, too. On the universal cover D~ of D one can introduce future sets in a canonical way. These sets are determined by the above cone field and by the choice of orientation in time; they form the causal structure on D~ (whereas D is acausal since it is compact).


as it is shown in [PaSe-82a]. Remark 1. In the General Relativity Theory (GRT), D~ is also known under the name of Einstein static universe (see [Kr-80, p.122]). The respective solution (of GRT Einstein equations) is interpreted as an ideal fluid. If not to assume R=1, then the scalar curvature is 6/(R2). Energy density and pressure both equal 1/(R2). Energy conditions hold. See [Le-07] for proofs. 2. The worlds L and F Topologically, L~ is R4. Its relatively compact form L (being a four-dimensional orbit in D) is determined by a basis of vector fields l1, l2, l3, l4 on U(2) where l1 = -(L-10 + L04 + L-11 + L14), l2 = (1/2)(L-12 + L24 + 2L30 + 2L31), l3 = (1/2)(L-13 + L34 + 2L02 + 2L12), l4 = (1/8)(-5L-10 - 3L-11 + 3L04 + 5L14 + 4L23). One can prove that they generate the oscillator Lie algebra. The expression for the invariant form (which determines the bi-invariant metric) follows from the formula for the respective wave operator (see below). The scalar curvature is now 0 (as shown in [Le86b] where this world L~ has been studied separately; altogether the three worlds have been studied in [Le-86a]). Here is the expression for the conformally covariant wave operator: 2l1l4 ­ (l2)2 - (l3)2. In terms of GRT, we now have an isotropic electromagnetic field determined by a covariantly constant light-like vector (see [Le-86b, p.123]). Energy conditions hold. This space-time is a special case of plane waves. The latter have been discussed in dozens of publications. Of special interest in our context is an article [Pe-76] by R. Penrose, where he introduced a method for taking a continuous limit of any space-time to a plane wave. In [NaWi-93] a conformal field theory model is based on L. The model is an un-gauged WessZumino-Witten model. In [CaJa-92], [CaJa-93] the oscillator Lie algebra l is used to formulate stringinspired lineal gravity as a gauge theory. However, the last two publications are seriously flawed. Namely, the upper left corner h (see expression (36) from [CaJa-92] and formula (3.41) from [CaJa-93]) has to be an identity matrix, rather than a diagonal matrix with 1, -1, entries. No surprise that the authors could not believe in one of their own conclusions (see their p. 249 of [CaJa-93]). In

[NaWi-93] (which refers to [CaJa-92], [CaJa-93]) the invariant form in question is introduced correctly: expression (6) on p.3751. The group L has been called an oscillator one. The L's important property to admit a nondegenerate bi-invariant metric has only been noticed in early 80s: [GuLe-84], [Le-85], [MeRe85]. From the above discussion it is clear that the building blocks D, L are quite well understood, and their importance (as specific worlds of the GRT and otherwise) is accepted by the physics/mathematics community. As part of the DLF-approach, let us now consider the tachionic component F. F~ is R4, topologically. It is the universal cover of the Lie group U(1,1). Its relatively compact form F (being a four-dimensional orbit in D) is determined by an orthonormal basis of vector fields H0, H1, H2, H3 on U(2). Here H0 = L-10 ­ L12, H1 = ­ L-12 ­ L01, H2 = L02 ­ L-11, H3 = L34. These fields generate a u(1,1), a sub-algebra of su(2,2). The scalar curvature is negative 6, and (H0)2 - (H1)2 - (H2)2 - (H3)2 ­ 1 is another conformally covariant wave operator. Remark 2. Treated as the solution of Einstein equations, it is interpreted as a tachionic fluid, [Kr80, p.57]. In the expression for the corresponding bi-invariant metric, there is a parameter a related to a choice of an invariant form on the simple su(1,1)sub-algebra of u(1,1). The scalar curvature is now 6/a2. Energy density and pressure are both negative; -1/(a2). These statements have been proven in [Le07]. The parameter a is a conformal invariant. Energy density and pressure both negative imply energy conditions violation, which is why the world F plays a special role. Here is what M. Davidson (an expert on tachyons) writes ([Da-01, p.1]): "Tachyons captured some interest in the physics community in the 1960s and 70s [1-7] (references from [Da-01] are not included into this short article; A.L.), but they have since fallen somewhat from fashion because direct experimental evidence has not been found to support their existence, and also because of concerns about causality [8]. Arguments have been made to counter the causality objections [9], and the issue remains in dispute. There are several reasons why tachyons are still of interest today, and in fact interest may be increasing. First, many string theories have tachyons occurring as some of the particles in the theory [10], although they are generally regarded as unphysical in those theories. There are also several recent papers that assert experimental evidence that some neutrinos are tachyons [11, 12]. There is a new and extensive reanalysis of tachyon dynamics [13]. There is much discussion in the physics literature in recent years


of superluminal connections implied by quantum mechanics and by the evanescent wave phenomenon of light optics as well as quantum tunneling, all indirect evidence of non-locality in nature. These and other recent developments show that tachyons are still a timely subject for investigation." 3. Pseudo-Hermitian realization of the Minkowski world M Let us first recall the well-known Hermitian model for M (see [PaSe­82a] or [Le-95]). Each event (or an element of M) is represented by a two by two Hermitian matrix h. The totality of all skew-Hermitian matrices ih forms a Lie algebra u(2). A typical element (t,L,f) of the simply connected eleven-dimensional (scaling included) Poincare group P~ maps h into etLhL* + f: h e LhL* + f
t

straightforward exercise to verify that the two actions commute with Q. Let us now introduce an analogue of the Caley map, CF, from u(1,1) into U(1,1): CF (h) = [1 ­ (shs)/2][1 + (shs)/2]
-1

(3.4).

Contrarily to the original Caley map CD , its analogue CF is not globally defined. As it follows from (5.4), the determinant of [1 + (shs)/2] vanishes on a certain (two-dimensional) hyperboloid of one sheet. That is why there is only local pseudo-Hermitian analogue of Theorem 2 from [Le-07]. More details are provided below. 4. F­represented SU(2,2) As part of the DLF-approach, consider the following matrix representation of the Lie group G = SU(2,2). It is conjugate to the D­representation (the latter has been originally introduced in Segal's Chronometry; see [PaSe-82a], or [Le-95]). That conjugation is performed by the following four-byfour matrix W: W is the direct sum of -1 with a certain three by three matrix. The only non-zero entries of the latter matrix are 1s on the auxiliary diagonal. Clearly, W2 equals the unit matrix. The D­represented SU(2,2)=G (call it DG, in brief) was composed of a certain set of pseudounitary matrices. Overall, DG has been defined with the help of a distinguished diagonal matrix, diag{1,1,-1,-1}. Under the conjugation by W we get S= diag {1,-1,-1,1}, which determines another copy of SU(2,2) (denote it by FG). Clearly, an isomorphism between DG and FG is carried out (via conjugation in SL(4,C)) by the matrix W. The group FG is composed of those matrices g (with unit determinant), which satisfy g*Sg=S (4.1).

(3.1).

In the above (3.1), t is a real number, L is a matrix from SL(2,C), f is a Hermitian matrix. It is a wellknown action of P~. The Caley map c =cD (which has been already mentioned in Section 1) is defined as follows: cD(h) = (1 + ih/2)(1 ­ ih/2)­1 (3.2).

The image of this map is an open dense subset of U(2). The group P~ acts on D=U(2), too. The Caley map intertwines respective actions (see Theorem 2 of [Le-07]). The possibility of the following pseudo-Hermitian picture seems to have been unnoticed. Recall that a two by two matrix h (with complex entries allowed) is in u(1,1) iff h*s + sh = 0, where s is a two by two matrix diag{1,-1}. Theorem 2. There is a linear bijection Q of the Lie algebra u(2) onto u(1,1), and there is such a P~action on u(1,1), that one gets a commutative diagram (in other words, Q intertwines respective P~-actions). Proof. Choose the bijection Q, which maps a Hermitian matrix

a

b

cd a - ic . The resulting Q is a bijection matrix - ib d
between two real four-dimensional subspaces in C4. If a matrix L is from SL(2,C), then it maps a pseudo-Hermitian matrix h into A*L'B*hALTB: h A*L'B*hALTB, (3.3)

into a pseudo-Hermitian

Similarly to the D­case, it is convenient to build each g of two-by-two blocks A, B, C, D. The maximal (essentially) compact subgroup K in D­ representation consisted of block-diagonal matrices g, that is, B=C=0. There is an analogue of K in Frepresentation, call it H. Formally, H is determined by the same condition as K was. Recall that the world F has been defined above as the Lie group U(1,1), see below, equipped with a certain biinvariant metric. The above matrix S is the following direct sum of two-by-two matrices: S = diag{s,-s}, where s = diag{1,-1}. Define U(1,1) as the totality of all two by two matrices satisfying

where L' is a complex conjugate of L (not a transpose), A = diag{1,i}, B = diag{-i,-1}, LTis the transpose of L. Scaling and parallel translations both act like before, see the law (3.1). It is a


z*sz=s

(4.2).

(N.Y.) 225(1993), 229-263 [Da-01] Mark Davidson,
http://www.arxiv.org/abs/quant-ph//abs/quantph/0103143v2>arXiv: quant-ph/0103143v2 downloadable from the physics archive at www.arxiv.org.

Lemma (it is an analogue of Lemma 2.1.4 from [PaSe-82a]). A matrix g from SL(4,C) belongs to FG if and only if the following conditions hold A*sA ­ C*sC = s, D*sD ­ B*sB = s, D*sC ­ B*sA = 0. (4.3) Based on (4.1) straightforward proof is omitted. Let us now introduce the following FG-action on F: an element g maps a matrix z into (Az + B)(Cz + D)-1: gz = (Az + B)(Cz + D)-1 (4.4).

Theorem 3. Equation (4.4) defines (formally) a left action on F=U(1,1), that is, (g'g)z = g'(gz). If the matrix Cz+D is non-degenerate, then gz belongs to F. The proof is omitted. Remark 3. It can be shown that for an arbitrarily chosen z from U(1,1), formula (4.4) is well-defined in a certain neighborhood of z, and for elements g from a certain neighborhood of a neutral element in FG. Such an action is called a local one. Here is an example when (4.4) is undefined. Take z with rows {21/2,1}, {1, 21/2}; take g determined by blocks A= D =

10 10 ,B=C= , 0 cht 0 sht

where ch t = 21/2, sh t = ­1, values of hyperbolic cosine and of hyperbolic sine. Theorem 4. Equation (4.4) defines a local FGaction on F = U(1,1). The subgroup H acts globally. The orbit of the neutral element (as well as the orbit of any other element of F) is the entire U(1,1). The proof is omitted. 5. Conclusions The main finding of the article is that the tachyonic component F can be introduced in a way similar to how the D-component has been treated in conventional physics. It is proposed that L- and Fcomponents of an object can play the role of (long wanted) hidden variables of quantum mechanics. References [Al-76] Alexandrov A.D. Vestnik LGU. Ser. Math., Mech., Astr. 19 (1976), no.4, 5-28 [CaJa-92] D. Cangemi and R. Jackiw, Phys. Rev. Lett., 69(1992), n.2, 233-236 [CaJa-93] D. Cangemi and R. Jackiw, Ann. Phys.

[DS-01] Daigneault A. And Sangalli A., Notices of the Amer.Math.Soc. 48 (2001), 9-16. [GuLe-84] Guts, Alexandr K., and Alexander V. Levichev, Doklady Akademii Nauk SSSR 277 (1984), 253-257; English transl. In Soviet Math. Dokl. 30(1984), 253-257. [K-02] Konstantin G. Korotkov. Human Energy Field: study with GDV bioelectrography.BACKBONE PUBLISHING Co., Fair Lawn, NJ, USA, 2002. [KL-05] K.Korotkov, A.Levichev; The 3-Fold Way and Consciousness Studies (2005), in the Electronic Library of the Institute of Time Nature Exploration, Moscow State University, http://www.chronos.msu [Kr-80] Kramer D., H. Stephani, M. MacCallum, E. Herlt, Exact Solutions of Einstein's Field Equations. VEB Deutscher Verlag der Wissenschaften, Berlin 1980. [Le-85] Levichev, A.V. Siberian Journal of Mathematics 26 (1985), n.5, 192-195 [Le-86a] Levichev, A.V., in "Group Theoretical Methods in Physics", Proc. of the III Intern. Sem., Yurmala, May 1985, vol.1 (M.A.Markov ed.), Nauka (1986), 145-150. [Le-86b] Levichev, A.V., Siberian Journal of Mathematics 27 (1986), 117-126 [Le-93] Levichev, A.V., Izvestia VUZov. Physics (1993), n.8, 84-89 [Le-95] Levichev A.V., ln: Semigroups in Algebra, Geometry, and Analysi, Eds. J.Hilgert, K.Hofmann, and J.Lawson, de Gryuter Expositions in Mathematics, Berlin 1995, viii+368 pp., 77-103, http://math.bu.edu/people/levit. [Le-03] Levichev, .V. Three symmetric worlds instead of the Minkowski space-time,Trans. RANS, ser. &C, 7 (2003), n.3-4, 87-93 [Le-07] Levichev, A.V., submitted to the Izvestia VUZov. Physics [MeRe-85] Medina A. and Ph. Revoy, Manuscripta Math. 52 (1985), 81-95 [NaWi-93] Chiara R. Nappi and Edward Witten, Phys. Rev. Lett., 71(1993), n.23, 3751-3753 [PaSe-82a] Paneitz, Stephen M., Irving E. Segal, Journal of Functional Analysis 47 (1982), 78-142 [Pe-76] R. Penrose, in Differential geometry and relativity, pp.271-275. Mathematical Phys. And Appl. Math., Vol. 3. Reidel, Dordrecht, 1976. [Se-76] Segal, Irving E., Mathematical Cosmology and Extragalactic Astronomy, Academic Press, New York, 1976. [Se-85] Segal, Irving E., in "Conference Proc., The Cosmic Background Radiation and Fundamental Physics," Publ. Soc. Ital. Fis. I (1985), 209-223.