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The characteristics and the relative conservation rules of the environment time-space with periodic symmetry
Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91 www.elsevier.com/locate/theochem
The characteristics and the relative conservation rules of the environment time-space with periodic symmetry Xuezhuang Zhaoa,*, Xizhang Yib, Daren Guanb, Xiufang Xua, Guichang Wanga, Zhenfeng Shanga, Zunsheng Caia, Yinming Pana a
b
Department of Chemistry, Nankai University, Tianjin 300071, People’s Republic of China Institute of Theoretical Chemistry, Shandong University, Jinan 250100, People’s Republic of China Received 7 January 2004; accepted 1 September 2004
Abstract In this work, we have put forward the conservation of generalized parity concerning the environment time-space, which involves the main points listed as follows: first, corresponding to the environment space with periodic symmetry the invariant is space-parity. Especially, for periodic translation the corresponding invariant is wave-parity. If the space periodic length approaches zero, the conservation of wave-parity will transform to the conservation of momentum. Second, corresponding to the environment time with periodic length the invariant is frequency-parity. If the time periodic length approaches zero, the conservation of frequency parity will transform to the conservation of energy. Third, for the environment time-space, if the time-space periodic length is very small but not null, we can get the formulae DEDtRh and DPDxRh, which are similar to the uncertainty principle. Finally, for an environment time-space with the measurable periodic length, we have analyzed the relative conservation rules. q 2004 Elsevier B.V. All rights reserved. Keywords: Conservation of generalized parity; Environment time-space; Periodic symmetry
1. Introduction To analyze the theoretic essentiality and the defects of the conservation principle of orbital and symmetry, i.e. the Woodward–Hoffmann (W–H) rule [1,2], we have probed into the chemical application of symmetry principle in field theory (Noether’s theorem) [3]. On this basis, we put forward The Conservation of Generalized Parity [4], by which, we can reveal the theoretic essentiality and point out some vague concepts in W–H rule. Meanwhile, we can deduce the W–H rule and other selection rules in chemistry on the common ground, and expand the W–H rule to more extensive range. In this paper, we will expand it further. Although such research is based on the symmetry principle in field theory (Noether’s theorem), the systems explored in field theory are in the four-dimensional (4D) * Corresponding author. Tel.: C86 22 2350 4854; fax: C86 22 2350 2458. E-mail address: [email protected] (X. Zhao). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2004.09.036
free time-space. However, in the chemistry systems, the relevant environment time-space is usually the non-free 3D space with a certain symmetry. In connection with these researches, we touch upon the environment space with the point group symmetry; and in connection with the study relevant to potential energy surface, we touch upon the socalled intrinsic reaction space with some special symmetry. In the discussion of solid physical processes, we touch upon the 3D environment space with the space group symmetry, from which we obtain the corresponding invariant, space parity. Obviously, the one-dimensional space group equals the space-translation group with the periodic symmetry in one-dimensional space. According to the relativistic theory, the time and space are interrelated, and the real 4D free time-space may be homogeneous and isotropic. However, the environment time-space suggested in this paper may not be so. In our previous work, we have inquired into some non-free environment space with some special symmetry, in particular, the 3D environment space provided with
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X. Zhao et al. / Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91
the translation periodic or near periodic symmetry. As for ‘time’, we are still considering it as free. Even for many natural time periodic phenomena in time, the time coordinate of the environment time-space is still treated as free time, in general. However, when we consider these periodic phenomena as how they affect a certain research object, the time in environment time-space may equally well be considered as no longer free, but we use the non-free environment time with the periodic translation symmetry. In such defined environment time-space, how to probe into the movement rules of the relative system will be discussed later in this paper. Providing the periodic or near periodic environment time-spaces, the relation phenomena are not rare, such as the movement of celestial body and the X-ray periodic pulse radiation of black hole in astronomy, the genetic and replication in biology, the stochastic resonance and forced oscillatory in non-linear dynamic physical or chemical processes, the microcosmic processes in relation to wave characteristic and so on. What conservation rules are determined by such symmetry? And how do these rules affect the development of nature? Undoubtedly, there are some important and perfect vistas in the forefront of these researches.
2. Results and discussion 2.1. The environment space with space group symmetry We had analyzed the movement rules of a system in the environment space with certain space group symmetry. For convenience sake, we sum up the main results related based on more foundational level here. As for further information, it may be found in Refs. [3,4]. For one-dimensional space (x) and time (t), the plane wave may be expressed as: Jðx; tÞ Z ð2pZÞK1=2 exp½ði=ZÞðpx K EtÞ Z ð2pZÞK1=2 exp½ikx K iut
(1)
where p and E are corresponding momentum and energy, respectively. k and u are the wave vector and angular frequency, respectively. Plank constant hZ2pZ. The corresponding field operators are [5]: X ^ tÞ Z Jðx; a^ ðk;uÞ exp½ikx K iut (2a) ðk;uÞ
^ Cðx; tÞ Z J
X
a^C ðk;uÞ exp½Kikx C iut
(2b)
ðk;uÞ
^ ðk;uÞ are the creation and annihilation where a^C ðk;uÞ and a operators corresponding to (k, u) state particle, respectively. In the environment space with one-dimensional periodic symmetry, as the periodic length equals ax, for one particle
system in k-state the state functional is expressed as: Fk ðrÞ Z a^C k F0
(3)
where a^C k and F0 are the creation operator of k-state system and the functional of vacuum state, respectively. According to Bloch’s theorem, as the periodic length ax and the ^ x ax Þ represents the translation of nxax transformation Tðn (where nx is an integer number), then for the eigenstate Fk(r) ^ x ax Þ, we can obtain of Tðn ^ x ax ÞFk ðrÞ Z exp½iknx ax Fk ðrÞ Tðn
(4)
^ x ax Þ with the eigenvalue i.e. Fk(r) is the eigenstate of Tðn (wave-parity): tðnx ax Þ Z exp½iknx ax
(5)
It is not difficult to prove that when k-state becomes the k 0 ZkC(2psx/ax)-state (where sx is a certain integer ^ x ax Þ number), Fk 0 (r) ought to be still the eigenstate of Tðn with the eigenvalue of t*(nxax)Zexp[iknxax] for any nx. In fact, k-state and kC(2psx/ax)-state represent the same state, i.e. k is denoted as a same number with the module (2p/ax), and only when ax is zero, k will be the common number with the module infinite. Due to the conservation of wave-parity, the k ought to be unchanged. When axZ0, k is common number, and the conservation of k will determine the conservation of momentum. It can be predicted that as the space translation periodic symmetry transform to the homogeneity of space, the conservation of wave-parity ought to transform to the conservation of momentum, p. These results in previous text may be spread to the threedimensional environment space. When the periodic length vectors are ax, ay, az, respectively, according to these vectors, there is a lattice environment space. We may analyze the space group in connection with this environment space. Corresponding to such space groups, the relative invariants are various generalized space-parities. In such environment space the particle movement-state may be described by means of the point in k-space constituted by reciprocal lattice. In this way, we may analyze the movement-rules of state change through some interaction processes in solid physics. 2.2. The environment time-space with time-periodic symmetry Now we turn to analyze the environment time-space with time-periodic symmetry. For such movement system, the relative potential function ought to be time-content obviously and it is a time periodic one. Because the time periodic symmetry group and one-dimensional space periodic symmetry group are isomorphic according to group theory in mathematics, then we can analyze the environment time-space with the time periodic symmetry as for the case of the environment space with one-dimensional space periodic symmetry, as what we have done.
X. Zhao et al. / Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91
For the environment time-space with the time-periodic length equal at, the state functional of u-state system Fu ðr; tÞ Z a^C u F0
(6)
where a^C u and F0 are the creation operator of u-state system and the functional of vacuum state, respectively. As the time periodic length is at and transformation T^ t ðnt at Þ represents the translation of ntat (where nt is an integer number) then about the eigenstate Fu(r,t) of T^ t ðnt atÞ, we can obtain: T^ t ðnt at ÞFu ðr; tÞ Z exp½iunt at Fu ðr; tÞ
(7)
i.e. Fu(r) is the eigenstate of T^ t ðnt at Þ with the eigenvalue (i.e. frequency-parity): tt ðnt atÞ
Z exp½iunt at
(8)
It is not difficult to prove that when u-state becomes the u 0 ZuC(2pst/at)-state (where st is a certain integer ^ t at Þ number), Fu 0 (r,t) ought to be still the eigenstate of Tðn with the eigenvalue of tt ðnt at ÞZ exp½iunt at for any nt. In fact, u-state and uC(2pst/at)- state may represent the same state, i.e. u may be denoted as the same number with the module (2p/at), and only if at is zero, u will be the common number with the module infinite. In our analysis of the onedimensional space periodic field, we introduce the reciprocal lattice space (k-space), using the 1/ax as the basic vector. Similarly, now we introduce the reciprocal periodical timeperiodic space (u-space, or angular frequency space) using ^ t at Þ and the 1/at as the basic vector. Since the operator Tðn ^ Hamiltonian H can be commuted, the eigenvalue, i.e. the the frequency-parity tt ðnt at Þ is a generalized parity and ought to be in conservation. As the conservation of frequencyparity, uZE/Z ought to be in conservation, too. If atZ0, u is a common number, and the conservation of u determines the conservation of energy, E. It can be predicated that as the time translation periodic symmetry transforms to the homogeneity of time, the conservation of frequency-parity ought to be transformed to the conservation of energy, E. 2.3. The environment time-space with shorter-periodic-length As mentioned above, in the case of the environment time-space with periodic symmetry, the corresponding frequency-parity and/or wave-parity ought to be in conservation. If the time-space periodic length is null, at Z 0 and ax Z 0
(9)
the periodic symmetry of time-space ought to be transformed to the homogeneity of time-space, meanwhile the corresponding conservation of wave-parity and frequencyparity will be transformed to the conservation of momentum and of energy, respectively. Now we turn to analyze the case with shorter periodic length, i.e. when at and ax are very small but not equal zero. First the case of periodic symmetry space is considered. As we have known, for k space,
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the k-state and kC(2psx/ax)-state here represent the same state (for any integer number sx), that is to say, the k-space is a periodic symmetry space with the periodic length (2p/ax). When k changes a measurement DkZ(2p/ax), the variation of momentum DpZDk-Zh/ax. It is notable that in this case Dp corresponds to one period in k-space (reciprocal lattice space). Since the non-homogeneity of space results from the space periodic symmetry, the measurement error Dx for xspace will introduce the undulation of momentum measure. If the periodic length ax of the environment space (i.e. the lattice space) is very small, the measurement error Dx will not be less than ax, i.e. DxRax. Thus, we have: Dp DxR Dp ax Z h
(10)
In the case of the time periodic symmetry, by means of the similar program, we can get: DE DtR DE at Z h
(11)
where DE corresponds to a period in u-space, Dt represents the measurement error of time and Dt will not be less than at, i.e. DtRat. These results are similar to the uncertainty principle. Is it an accidental coincidence only, or an inevitable intrinsic character? And can the time-space quantization be understood in this way? It ought to be noticed! Perhaps, it means that if the periodic length of the environment (maybe, even real) time-space is too small (less than the measurement error), corresponding unhomogeneity of time-space from the periodic symmetry, the variant of momentum and energy will be covered by the measurement error and could not be detected. And then we can consider such time-space as the homogeneous free one. 2.4. The environment time-space with longer-periodic-length Now we analyze the environment time-space with the measurable periodic length. As for such environment space, i.e. the environment space with the space-group symmetry, we have somewhat analyzed them already [4,5]. In such case, the related invariant is space-parity (a kind of generalized parity) and would comply with the conservation rules of space parity. According to these rules we may explain some relative processes of solid physics correctly, such as the electron transition process of semi-conductor from the occupied band to the conduct band. Now we will analyze the time periodic length in measurable region. The corresponding movement trajectory in phase-space ought to be the closed curve, such as the limit-cycle. There are many researches about the invariants of the limit-cycle movement, especially in the study of nonlinear kinetics in physics and chemistry, but we take another look at it from a different angle. For convenience, we introduce Tt-[at] to represent the time periodic symmetry group with the time periodic length at. Obviously, Tt-[2a] is one subgroup of Tt-[a].
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Since symmetry transformation T^ t ð2aÞ would be contained in both Tt-[2a] and Tt-[a], while T^ t ðaÞ will be contained in Tt-[a] only, but not in Tt-[2a]. Then we can use the Tt-[0] to represent the symmetry group of time homogeneity. For any a, all Tt-[a] are the subgroup of Tt[0]. It means that for any periodic length (a) the corresponding symmetry group ought to be the subgroup of time-homogeneity. Generally speaking, the bigger a is, the poorer the symmetry of Tt-[a] would be. Consider an oscillation system in free space. For this system there is the Tt-[ai] symmetry and the intrinsic angular frequency uiZ2p/ai, as if there is another external field with the Tt-[at] symmetry, (the forced oscillation with the angular frequency ut), then it will form a final state with the Tt[af] symmetry (corresponding final state, the angular frequency is uf). In this process, its frequency-parity and the relative angular frequency, u ought to be invariable. It is notable that the so-called angular frequency would correspond to the symmetry of whole process. Corresponding symmetry group ought to be the common subgroup Tt-[a0] of Tt-[ai], Tt-[af] and Tt-[at]. In most cases, the symmetry of Tt-[af] and Tt-[a0] are the same. In general, the condition is that the a0 will be the common multiple of ai, af and at. If a0 becomes bigger, the opportunity to realize such process will be less. If ai is null, that means the initial state is invariable for time, i.e. the stable state or the equilibrium state. In the external field with the Tt-[at] symmetry, the finial forced oscillation state ought to be the Tt-[at] symmetry, i.e. afZat, such as in the action of various ultrasonic field on the relaxation of equilibrium-state, the phases and amplitudes of the final relaxation oscillations, but the oscillation periodic length (or frequency) will be the same as in ultrasonic field [6]. As for the noise oscillation field, it will be variable vs the time. However, such variable is stochastic, and sometimes we may consider it as the Tt-[0] symmetry. As an example, for a certain bistable state system, when it is in one stable state of this bistable state, even if there is some noise oscillation, we may consider it as holding the near Tt-[0] symmetry. When it is forced by an oscillation with the Tt-[at] symmetry, the stochastic resonance may arise, and this system will be provided the Tt-[af]Znear Tt-[at] symmetry [7]. If all of a0, ai, af and at, are the same, the probability will be the biggest and that means the resonance. Under the condition of the invariable ai and at (not equal to zero), for different amplitude of at, the af may be variable [8], while the conservation condition are satisfied still. For example, in the initial inherent oscillation system with the initial symmetry Tt-[ai], by means of the forced oscillation with the Tt-[at] symmetry to change the environment time-space and to form the final oscillation state with the Tt-[af] symmetry. For such oscillation, in general, it may decompose as the overlap of two oscillations with the Tt-[ai] symmetry and the Tt[at] symmetry, respectively, by means of the Fourier
transformation analysis. Especially, when at is integral (n) fold of ai: Tt-[a0]ZTt-[at]ZTt-[nai]. It is notable that the equal sign denotes the oscillation symmetry (or period) ought to be the same, but not the oscillate waveshape. It means corresponding oscillation will be provided with the same time periodic length, but the wave shapes will be repeated, respectively. Meanwhile, it is ought to be notable too that the initial inherent oscillation state usually bears both Tt-[ai] and Tt-[nai] symmetry. However, in general, the final state provide the Tt-[af]ZTt-[a0]ZTt-[nai] symmetry only, but not the Tt-[ai] symmetry. In more general case, the a0 is the common multiple of ai and at, a0ZniaiZntat then the final state will hold the Tt-[af]ZTt-[a0]ZTt-[niai]ZTt[ntat] symmetry, where ni and nt are corresponding integral number. If the symmetry of Tt-[ai] is less than that of Tt-[at], i.e. aiZntat (ntO1), the system will conserve the Tt-[ai] symmetry. Especially, when the action of at is poor (the corresponding amplitude is less) the relative effect is often ambiguous. We may find some examples in such case [8]. Some of our theoretical and experimental researches on BZ reaction may be an illustration of such case [9]. It ought to be noticed that corresponding to a certain symmetry Tt-[a] there are some various states. It is the same as a certain k space corresponding to a certain space group (lattice) space and the various points in this k space will be represented in different states. For the various points in a certain u-space, the corresponding system would be in the same symmetry, their periods, at, are the same, while the frequency-parities (and then the u) are different. Also the frequency-parities (and then the u) would be related to the oscillation phase of various systems. When the disparity of u equals 2pst/at (st is a certain integer number), corresponding frequency-parity ought to be invariable. Their (relative) oscillation phases are the same, so we can discuss the variation rule of oscillation phase by means of the conservation of frequency-parity. For a certain single system, if its period (or frequency) is invariable, i.e. corresponding symmetry is unchanging, its oscillation phase (related to a certain standard state) is obviously invariable. For the interaction systems with the same symmetry but different oscillation phases, the final oscillation phase ought to conform to the conservation condition of frequencyparity, then we can probe into the corresponding variation rules of oscillation phases. As mentioned above, we have analyzed the cases of time and space provided the periodic symmetry, respectively. For both time and space provided the periodic symmetry, they often appear in physical and/or chemical nonlinear dynamic system in connection with the diffusion and flow, and could be analyzed by means of the corresponding conservation rules of generalized parity.
X. Zhao et al. / Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91
3. Conclusions In this paper, the symmetry and conservation rules in connection with the environment time-space with periodic symmetry are analyzed. The main points of this paper are as follows: (1) Having summarized our previous work in relation to the environment space with periodic symmetry and the conservation rules. To point out corresponding to the space-group symmetry the invariant is space-parity. Especially, for periodic translation the corresponding invariant is wave-parity. The eigenstates of wave-parity correspond to the wave vector in k-space in solid physics. If the space periodic length approaches zero, the conservation of wave-parity may transform to the conservation of momentum. (2) Using the similar method, we may analyze the environment time with periodic length. For such symmetry, the corresponding invariant is frequency parity. As similarly relate the k-space in solid physics, we may construct a u-space (generalized) with the reciprocal time periodic length as the basis. The eigenstate of frequency-parity relates a vector in uspace. If the time-periodic length approaches zero, the conservation of frequency parity may transform to the conservation of energy. (3) For the environment time-space, if the time-space periodic length is very small but not null, we can get the formula similar to the uncertainty principle. According to the relative result, we may obtain the suggestion that there is a probable experimental definition for a time-space transformed from a periodic symmetry to homogeneity. (4) For an environment time-space with the measurable periodic length, we analyze the relative conservation rules. For periodic space in connection with the space symmetry group, we have studied the relative invariant and selection rules for relative processes in our previous
91
papers. In this paper, we study relative topic for the time-periodic symmetry further. Some regular results in connection with the forced oscillation in nonlinear dynamics are analyzed. It ought to be noted that the condition of the environment time-space with periodic or quasi-periodic symmetry is not rare in nature. Although the exploration in this paper is an important field, it is only a beginning. There is a fertile virgin land waiting to be opened up, and there are many work ought to be analyzed further. Although for our research on generalized parity the main point is the conservation rules in chemistry, it may be expected to expand to other scientific categories such as biochemistry, biological science, celestial science and so on. Therefore we ought to analyze the new kinds of symmetry. That is the purpose of this paper, too, and we are going to analyze some systems with the other new kinds of symmetry.
References [1] R.B. Woodward, R. Hoffmann, Angew. Chem. 8 (1969) 781. [2] R.B. Woodward, R. Hoffmann, The Conservation of Orbital Symmetry, Weinheis 1970. [3] X. Zhao, Application of Symmetry Principle in Field Theory to Chemistry, Science Press, Beijing, PRC, 1986 (in Chinese), CA 106:B38793f. [4] X. Zhao, Z. Cai, G. Wang, Y. Pan, B. Wu, J. Mol. Struct. (Theochem). 586 (2002) 209. [5] S.S. Scbweber, An Introduction to Relativistic Quantum Field Theory. Elmsford, New York 1961. [6] F. Wilkinson, Chemical Kinetics and Reaction Mechanisms, Van Nostrand Reinhold, New York, 1980. [7] G. Hu, Stochastic Forces and Nonlinear Systems, Shanghai Scientific and Technological Education Publishing House, 1995 (in Chinese). [8] K.S. Scott, Chemical Chaos, Clarendon Press, Oxford, 1991. [9] Q. Wei, Z. Hou, L. Chen, Z. Cai, X. Zhao, Acta Physico-Chemica Sinica, PRC 16 (2000) 338.
The characteristics and the relative conservation rules of the environment time-space with periodic symmetry Xuezhuang Zhaoa,*, Xizhang Yib, Daren Guanb, Xiufang Xua, Guichang Wanga, Zhenfeng Shanga, Zunsheng Caia, Yinming Pana a
b
Department of Chemistry, Nankai University, Tianjin 300071, People’s Republic of China Institute of Theoretical Chemistry, Shandong University, Jinan 250100, People’s Republic of China Received 7 January 2004; accepted 1 September 2004
Abstract In this work, we have put forward the conservation of generalized parity concerning the environment time-space, which involves the main points listed as follows: first, corresponding to the environment space with periodic symmetry the invariant is space-parity. Especially, for periodic translation the corresponding invariant is wave-parity. If the space periodic length approaches zero, the conservation of wave-parity will transform to the conservation of momentum. Second, corresponding to the environment time with periodic length the invariant is frequency-parity. If the time periodic length approaches zero, the conservation of frequency parity will transform to the conservation of energy. Third, for the environment time-space, if the time-space periodic length is very small but not null, we can get the formulae DEDtRh and DPDxRh, which are similar to the uncertainty principle. Finally, for an environment time-space with the measurable periodic length, we have analyzed the relative conservation rules. q 2004 Elsevier B.V. All rights reserved. Keywords: Conservation of generalized parity; Environment time-space; Periodic symmetry
1. Introduction To analyze the theoretic essentiality and the defects of the conservation principle of orbital and symmetry, i.e. the Woodward–Hoffmann (W–H) rule [1,2], we have probed into the chemical application of symmetry principle in field theory (Noether’s theorem) [3]. On this basis, we put forward The Conservation of Generalized Parity [4], by which, we can reveal the theoretic essentiality and point out some vague concepts in W–H rule. Meanwhile, we can deduce the W–H rule and other selection rules in chemistry on the common ground, and expand the W–H rule to more extensive range. In this paper, we will expand it further. Although such research is based on the symmetry principle in field theory (Noether’s theorem), the systems explored in field theory are in the four-dimensional (4D) * Corresponding author. Tel.: C86 22 2350 4854; fax: C86 22 2350 2458. E-mail address: [email protected] (X. Zhao). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2004.09.036
free time-space. However, in the chemistry systems, the relevant environment time-space is usually the non-free 3D space with a certain symmetry. In connection with these researches, we touch upon the environment space with the point group symmetry; and in connection with the study relevant to potential energy surface, we touch upon the socalled intrinsic reaction space with some special symmetry. In the discussion of solid physical processes, we touch upon the 3D environment space with the space group symmetry, from which we obtain the corresponding invariant, space parity. Obviously, the one-dimensional space group equals the space-translation group with the periodic symmetry in one-dimensional space. According to the relativistic theory, the time and space are interrelated, and the real 4D free time-space may be homogeneous and isotropic. However, the environment time-space suggested in this paper may not be so. In our previous work, we have inquired into some non-free environment space with some special symmetry, in particular, the 3D environment space provided with
88
X. Zhao et al. / Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91
the translation periodic or near periodic symmetry. As for ‘time’, we are still considering it as free. Even for many natural time periodic phenomena in time, the time coordinate of the environment time-space is still treated as free time, in general. However, when we consider these periodic phenomena as how they affect a certain research object, the time in environment time-space may equally well be considered as no longer free, but we use the non-free environment time with the periodic translation symmetry. In such defined environment time-space, how to probe into the movement rules of the relative system will be discussed later in this paper. Providing the periodic or near periodic environment time-spaces, the relation phenomena are not rare, such as the movement of celestial body and the X-ray periodic pulse radiation of black hole in astronomy, the genetic and replication in biology, the stochastic resonance and forced oscillatory in non-linear dynamic physical or chemical processes, the microcosmic processes in relation to wave characteristic and so on. What conservation rules are determined by such symmetry? And how do these rules affect the development of nature? Undoubtedly, there are some important and perfect vistas in the forefront of these researches.
2. Results and discussion 2.1. The environment space with space group symmetry We had analyzed the movement rules of a system in the environment space with certain space group symmetry. For convenience sake, we sum up the main results related based on more foundational level here. As for further information, it may be found in Refs. [3,4]. For one-dimensional space (x) and time (t), the plane wave may be expressed as: Jðx; tÞ Z ð2pZÞK1=2 exp½ði=ZÞðpx K EtÞ Z ð2pZÞK1=2 exp½ikx K iut
(1)
where p and E are corresponding momentum and energy, respectively. k and u are the wave vector and angular frequency, respectively. Plank constant hZ2pZ. The corresponding field operators are [5]: X ^ tÞ Z Jðx; a^ ðk;uÞ exp½ikx K iut (2a) ðk;uÞ
^ Cðx; tÞ Z J
X
a^C ðk;uÞ exp½Kikx C iut
(2b)
ðk;uÞ
^ ðk;uÞ are the creation and annihilation where a^C ðk;uÞ and a operators corresponding to (k, u) state particle, respectively. In the environment space with one-dimensional periodic symmetry, as the periodic length equals ax, for one particle
system in k-state the state functional is expressed as: Fk ðrÞ Z a^C k F0
(3)
where a^C k and F0 are the creation operator of k-state system and the functional of vacuum state, respectively. According to Bloch’s theorem, as the periodic length ax and the ^ x ax Þ represents the translation of nxax transformation Tðn (where nx is an integer number), then for the eigenstate Fk(r) ^ x ax Þ, we can obtain of Tðn ^ x ax ÞFk ðrÞ Z exp½iknx ax Fk ðrÞ Tðn
(4)
^ x ax Þ with the eigenvalue i.e. Fk(r) is the eigenstate of Tðn (wave-parity): tðnx ax Þ Z exp½iknx ax
(5)
It is not difficult to prove that when k-state becomes the k 0 ZkC(2psx/ax)-state (where sx is a certain integer ^ x ax Þ number), Fk 0 (r) ought to be still the eigenstate of Tðn with the eigenvalue of t*(nxax)Zexp[iknxax] for any nx. In fact, k-state and kC(2psx/ax)-state represent the same state, i.e. k is denoted as a same number with the module (2p/ax), and only when ax is zero, k will be the common number with the module infinite. Due to the conservation of wave-parity, the k ought to be unchanged. When axZ0, k is common number, and the conservation of k will determine the conservation of momentum. It can be predicted that as the space translation periodic symmetry transform to the homogeneity of space, the conservation of wave-parity ought to transform to the conservation of momentum, p. These results in previous text may be spread to the threedimensional environment space. When the periodic length vectors are ax, ay, az, respectively, according to these vectors, there is a lattice environment space. We may analyze the space group in connection with this environment space. Corresponding to such space groups, the relative invariants are various generalized space-parities. In such environment space the particle movement-state may be described by means of the point in k-space constituted by reciprocal lattice. In this way, we may analyze the movement-rules of state change through some interaction processes in solid physics. 2.2. The environment time-space with time-periodic symmetry Now we turn to analyze the environment time-space with time-periodic symmetry. For such movement system, the relative potential function ought to be time-content obviously and it is a time periodic one. Because the time periodic symmetry group and one-dimensional space periodic symmetry group are isomorphic according to group theory in mathematics, then we can analyze the environment time-space with the time periodic symmetry as for the case of the environment space with one-dimensional space periodic symmetry, as what we have done.
X. Zhao et al. / Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91
For the environment time-space with the time-periodic length equal at, the state functional of u-state system Fu ðr; tÞ Z a^C u F0
(6)
where a^C u and F0 are the creation operator of u-state system and the functional of vacuum state, respectively. As the time periodic length is at and transformation T^ t ðnt at Þ represents the translation of ntat (where nt is an integer number) then about the eigenstate Fu(r,t) of T^ t ðnt atÞ, we can obtain: T^ t ðnt at ÞFu ðr; tÞ Z exp½iunt at Fu ðr; tÞ
(7)
i.e. Fu(r) is the eigenstate of T^ t ðnt at Þ with the eigenvalue (i.e. frequency-parity): tt ðnt atÞ
Z exp½iunt at
(8)
It is not difficult to prove that when u-state becomes the u 0 ZuC(2pst/at)-state (where st is a certain integer ^ t at Þ number), Fu 0 (r,t) ought to be still the eigenstate of Tðn with the eigenvalue of tt ðnt at ÞZ exp½iunt at for any nt. In fact, u-state and uC(2pst/at)- state may represent the same state, i.e. u may be denoted as the same number with the module (2p/at), and only if at is zero, u will be the common number with the module infinite. In our analysis of the onedimensional space periodic field, we introduce the reciprocal lattice space (k-space), using the 1/ax as the basic vector. Similarly, now we introduce the reciprocal periodical timeperiodic space (u-space, or angular frequency space) using ^ t at Þ and the 1/at as the basic vector. Since the operator Tðn ^ Hamiltonian H can be commuted, the eigenvalue, i.e. the the frequency-parity tt ðnt at Þ is a generalized parity and ought to be in conservation. As the conservation of frequencyparity, uZE/Z ought to be in conservation, too. If atZ0, u is a common number, and the conservation of u determines the conservation of energy, E. It can be predicated that as the time translation periodic symmetry transforms to the homogeneity of time, the conservation of frequency-parity ought to be transformed to the conservation of energy, E. 2.3. The environment time-space with shorter-periodic-length As mentioned above, in the case of the environment time-space with periodic symmetry, the corresponding frequency-parity and/or wave-parity ought to be in conservation. If the time-space periodic length is null, at Z 0 and ax Z 0
(9)
the periodic symmetry of time-space ought to be transformed to the homogeneity of time-space, meanwhile the corresponding conservation of wave-parity and frequencyparity will be transformed to the conservation of momentum and of energy, respectively. Now we turn to analyze the case with shorter periodic length, i.e. when at and ax are very small but not equal zero. First the case of periodic symmetry space is considered. As we have known, for k space,
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the k-state and kC(2psx/ax)-state here represent the same state (for any integer number sx), that is to say, the k-space is a periodic symmetry space with the periodic length (2p/ax). When k changes a measurement DkZ(2p/ax), the variation of momentum DpZDk-Zh/ax. It is notable that in this case Dp corresponds to one period in k-space (reciprocal lattice space). Since the non-homogeneity of space results from the space periodic symmetry, the measurement error Dx for xspace will introduce the undulation of momentum measure. If the periodic length ax of the environment space (i.e. the lattice space) is very small, the measurement error Dx will not be less than ax, i.e. DxRax. Thus, we have: Dp DxR Dp ax Z h
(10)
In the case of the time periodic symmetry, by means of the similar program, we can get: DE DtR DE at Z h
(11)
where DE corresponds to a period in u-space, Dt represents the measurement error of time and Dt will not be less than at, i.e. DtRat. These results are similar to the uncertainty principle. Is it an accidental coincidence only, or an inevitable intrinsic character? And can the time-space quantization be understood in this way? It ought to be noticed! Perhaps, it means that if the periodic length of the environment (maybe, even real) time-space is too small (less than the measurement error), corresponding unhomogeneity of time-space from the periodic symmetry, the variant of momentum and energy will be covered by the measurement error and could not be detected. And then we can consider such time-space as the homogeneous free one. 2.4. The environment time-space with longer-periodic-length Now we analyze the environment time-space with the measurable periodic length. As for such environment space, i.e. the environment space with the space-group symmetry, we have somewhat analyzed them already [4,5]. In such case, the related invariant is space-parity (a kind of generalized parity) and would comply with the conservation rules of space parity. According to these rules we may explain some relative processes of solid physics correctly, such as the electron transition process of semi-conductor from the occupied band to the conduct band. Now we will analyze the time periodic length in measurable region. The corresponding movement trajectory in phase-space ought to be the closed curve, such as the limit-cycle. There are many researches about the invariants of the limit-cycle movement, especially in the study of nonlinear kinetics in physics and chemistry, but we take another look at it from a different angle. For convenience, we introduce Tt-[at] to represent the time periodic symmetry group with the time periodic length at. Obviously, Tt-[2a] is one subgroup of Tt-[a].
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Since symmetry transformation T^ t ð2aÞ would be contained in both Tt-[2a] and Tt-[a], while T^ t ðaÞ will be contained in Tt-[a] only, but not in Tt-[2a]. Then we can use the Tt-[0] to represent the symmetry group of time homogeneity. For any a, all Tt-[a] are the subgroup of Tt[0]. It means that for any periodic length (a) the corresponding symmetry group ought to be the subgroup of time-homogeneity. Generally speaking, the bigger a is, the poorer the symmetry of Tt-[a] would be. Consider an oscillation system in free space. For this system there is the Tt-[ai] symmetry and the intrinsic angular frequency uiZ2p/ai, as if there is another external field with the Tt-[at] symmetry, (the forced oscillation with the angular frequency ut), then it will form a final state with the Tt[af] symmetry (corresponding final state, the angular frequency is uf). In this process, its frequency-parity and the relative angular frequency, u ought to be invariable. It is notable that the so-called angular frequency would correspond to the symmetry of whole process. Corresponding symmetry group ought to be the common subgroup Tt-[a0] of Tt-[ai], Tt-[af] and Tt-[at]. In most cases, the symmetry of Tt-[af] and Tt-[a0] are the same. In general, the condition is that the a0 will be the common multiple of ai, af and at. If a0 becomes bigger, the opportunity to realize such process will be less. If ai is null, that means the initial state is invariable for time, i.e. the stable state or the equilibrium state. In the external field with the Tt-[at] symmetry, the finial forced oscillation state ought to be the Tt-[at] symmetry, i.e. afZat, such as in the action of various ultrasonic field on the relaxation of equilibrium-state, the phases and amplitudes of the final relaxation oscillations, but the oscillation periodic length (or frequency) will be the same as in ultrasonic field [6]. As for the noise oscillation field, it will be variable vs the time. However, such variable is stochastic, and sometimes we may consider it as the Tt-[0] symmetry. As an example, for a certain bistable state system, when it is in one stable state of this bistable state, even if there is some noise oscillation, we may consider it as holding the near Tt-[0] symmetry. When it is forced by an oscillation with the Tt-[at] symmetry, the stochastic resonance may arise, and this system will be provided the Tt-[af]Znear Tt-[at] symmetry [7]. If all of a0, ai, af and at, are the same, the probability will be the biggest and that means the resonance. Under the condition of the invariable ai and at (not equal to zero), for different amplitude of at, the af may be variable [8], while the conservation condition are satisfied still. For example, in the initial inherent oscillation system with the initial symmetry Tt-[ai], by means of the forced oscillation with the Tt-[at] symmetry to change the environment time-space and to form the final oscillation state with the Tt-[af] symmetry. For such oscillation, in general, it may decompose as the overlap of two oscillations with the Tt-[ai] symmetry and the Tt[at] symmetry, respectively, by means of the Fourier
transformation analysis. Especially, when at is integral (n) fold of ai: Tt-[a0]ZTt-[at]ZTt-[nai]. It is notable that the equal sign denotes the oscillation symmetry (or period) ought to be the same, but not the oscillate waveshape. It means corresponding oscillation will be provided with the same time periodic length, but the wave shapes will be repeated, respectively. Meanwhile, it is ought to be notable too that the initial inherent oscillation state usually bears both Tt-[ai] and Tt-[nai] symmetry. However, in general, the final state provide the Tt-[af]ZTt-[a0]ZTt-[nai] symmetry only, but not the Tt-[ai] symmetry. In more general case, the a0 is the common multiple of ai and at, a0ZniaiZntat then the final state will hold the Tt-[af]ZTt-[a0]ZTt-[niai]ZTt[ntat] symmetry, where ni and nt are corresponding integral number. If the symmetry of Tt-[ai] is less than that of Tt-[at], i.e. aiZntat (ntO1), the system will conserve the Tt-[ai] symmetry. Especially, when the action of at is poor (the corresponding amplitude is less) the relative effect is often ambiguous. We may find some examples in such case [8]. Some of our theoretical and experimental researches on BZ reaction may be an illustration of such case [9]. It ought to be noticed that corresponding to a certain symmetry Tt-[a] there are some various states. It is the same as a certain k space corresponding to a certain space group (lattice) space and the various points in this k space will be represented in different states. For the various points in a certain u-space, the corresponding system would be in the same symmetry, their periods, at, are the same, while the frequency-parities (and then the u) are different. Also the frequency-parities (and then the u) would be related to the oscillation phase of various systems. When the disparity of u equals 2pst/at (st is a certain integer number), corresponding frequency-parity ought to be invariable. Their (relative) oscillation phases are the same, so we can discuss the variation rule of oscillation phase by means of the conservation of frequency-parity. For a certain single system, if its period (or frequency) is invariable, i.e. corresponding symmetry is unchanging, its oscillation phase (related to a certain standard state) is obviously invariable. For the interaction systems with the same symmetry but different oscillation phases, the final oscillation phase ought to conform to the conservation condition of frequencyparity, then we can probe into the corresponding variation rules of oscillation phases. As mentioned above, we have analyzed the cases of time and space provided the periodic symmetry, respectively. For both time and space provided the periodic symmetry, they often appear in physical and/or chemical nonlinear dynamic system in connection with the diffusion and flow, and could be analyzed by means of the corresponding conservation rules of generalized parity.
X. Zhao et al. / Journal of Molecular Structure: THEOCHEM 713 (2005) 87–91
3. Conclusions In this paper, the symmetry and conservation rules in connection with the environment time-space with periodic symmetry are analyzed. The main points of this paper are as follows: (1) Having summarized our previous work in relation to the environment space with periodic symmetry and the conservation rules. To point out corresponding to the space-group symmetry the invariant is space-parity. Especially, for periodic translation the corresponding invariant is wave-parity. The eigenstates of wave-parity correspond to the wave vector in k-space in solid physics. If the space periodic length approaches zero, the conservation of wave-parity may transform to the conservation of momentum. (2) Using the similar method, we may analyze the environment time with periodic length. For such symmetry, the corresponding invariant is frequency parity. As similarly relate the k-space in solid physics, we may construct a u-space (generalized) with the reciprocal time periodic length as the basis. The eigenstate of frequency-parity relates a vector in uspace. If the time-periodic length approaches zero, the conservation of frequency parity may transform to the conservation of energy. (3) For the environment time-space, if the time-space periodic length is very small but not null, we can get the formula similar to the uncertainty principle. According to the relative result, we may obtain the suggestion that there is a probable experimental definition for a time-space transformed from a periodic symmetry to homogeneity. (4) For an environment time-space with the measurable periodic length, we analyze the relative conservation rules. For periodic space in connection with the space symmetry group, we have studied the relative invariant and selection rules for relative processes in our previous
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papers. In this paper, we study relative topic for the time-periodic symmetry further. Some regular results in connection with the forced oscillation in nonlinear dynamics are analyzed. It ought to be noted that the condition of the environment time-space with periodic or quasi-periodic symmetry is not rare in nature. Although the exploration in this paper is an important field, it is only a beginning. There is a fertile virgin land waiting to be opened up, and there are many work ought to be analyzed further. Although for our research on generalized parity the main point is the conservation rules in chemistry, it may be expected to expand to other scientific categories such as biochemistry, biological science, celestial science and so on. Therefore we ought to analyze the new kinds of symmetry. That is the purpose of this paper, too, and we are going to analyze some systems with the other new kinds of symmetry.
References [1] R.B. Woodward, R. Hoffmann, Angew. Chem. 8 (1969) 781. [2] R.B. Woodward, R. Hoffmann, The Conservation of Orbital Symmetry, Weinheis 1970. [3] X. Zhao, Application of Symmetry Principle in Field Theory to Chemistry, Science Press, Beijing, PRC, 1986 (in Chinese), CA 106:B38793f. [4] X. Zhao, Z. Cai, G. Wang, Y. Pan, B. Wu, J. Mol. Struct. (Theochem). 586 (2002) 209. [5] S.S. Scbweber, An Introduction to Relativistic Quantum Field Theory. Elmsford, New York 1961. [6] F. Wilkinson, Chemical Kinetics and Reaction Mechanisms, Van Nostrand Reinhold, New York, 1980. [7] G. Hu, Stochastic Forces and Nonlinear Systems, Shanghai Scientific and Technological Education Publishing House, 1995 (in Chinese). [8] K.S. Scott, Chemical Chaos, Clarendon Press, Oxford, 1991. [9] Q. Wei, Z. Hou, L. Chen, Z. Cai, X. Zhao, Acta Physico-Chemica Sinica, PRC 16 (2000) 338.