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The Bethe lattice and the dimension of micro spacetime
Pergamon
The Bethe
Lattice
and the Dimension
of Micro
Spacetime
M. S. EL NASCHIE Department of Applied Mathematics and Theoretical Physics, tiniversity of Cambridge, Silver Street. Cambridge. UK Abstract-An analogy is proposed between the Bethe lattice and the Cantorian spacetime picture of quantum physics, showing that in both cases the Hausdorff dimension of an effectively quasi-fractal meanfiled theory is equal to 4. This is identical to the upper limit of the Hausdorff dimension of a branched polymer, namely 2MI(M - I), for M = 2.0 1997 Elsevier Science Ltd
In this short note, we shall be concerned with an instructive analogy between the expectation value of the dimension of micro spacetime and a class of hierarchical lattices known as Bethe lattices $8 or simply Cayley tree [l, 21. Micro spacetime %(J-j is assumed to be an infinitedimensional transfinite Cantorian manifold [3-51. Somewhat similar to CFin dimensionality (n = m) and its geometry, the s lattice is used widely to model branched polymers and is frequently the only way to an exact solution [l, 21. Our basic idea in the present note is, in effect, to combine several of the relatively recent results obtained in the context of quantum diffusion [6] and quantum physics in fractalCantorian spacetime setting 13-51 and then show that both %‘-) and % must have the same Hausdorff dimension, namely dH = dB equal to 4. To do this, we need to recall the following results: 1. It is well-known from the work of Feynman [7] and Ord and others [3-51 that the path of a quantum particle is in general non-differentiable and fractal-like. It has been further established by Abbott et al. [8], Ord, Nottale and others that this fractal path must have a Hausdorff dimension d, = 2 irrespective of the dimension of the embedding space [9] which can be infinite. 2. There is a classical analogy between the diffusion equation and the Schrodinger equation that has been utilized by many researchers using different techniques such as analytical continuation to find a ‘realistic’ interpretation of orthodox quantum mechanics [6, 3-51. 3. It is by now quite well understood through the work of Nagasawa [6] and others, using the adjoint system method [lo], that a Schriidinger equation may be replaced by two diffusion equations in duality. In this way, Nagasawa was able to show that the Born interpretation P = (94) is more or less a derivable formula [6, lo] and an integral part of the equations of quantum mechanics. 4. Using the adjoint system method, the writer and others were able to show that the Born formula can be interpreted as a quantum probability density defined by a pair of conjugate complex Schriidinger equations which may be transformed to a pair of diffusion equations using complex time [5]. This particular interpretation is in complete harmony with (a) Eddington-Schrodinger interpretation [9], (b) Cramer’s transactional interpretation af quantum mechanical measurement [9, 111 and (c) the Wheeler-Feynman absorber theory [9-111. Combining all the results and ideas of the preceding four points, we come to the following quantum spacetime picture and the corresponding 3 lattice analogy. 1887
M. S. EL NASCHIE
1888
To develop this picture, we must first attempt to imagine a fractal path representing in effect a fractal geodesic in n-dimensional spacetime. Next, and in analogy to complex time [5] and Nagasawa’s dual diffusion [6], we imagine that we are superimposing transversally on this geodesic a second one which is supposed to be ‘running’ back in time and is thus a quasi ‘antigeodesic’, to use our earlier terminology for want of a better one. Thus any real measurement on EcX) is bound now to be taken on geodesic ‘points’ formed by the intersection or union of the two sets forming the two transversal geodesics. This will immediately imply a critical Hausdorff dimension dH equal to (Dim,r\%(“‘)=
~~=(a,=*)O(;,=2)=‘d),O’dl..
(1)
or (Dim,vS(n,)=~~=(d==2)0(;7,=2)=‘I;),0’hl,
(2)
That means that direct products and direct sums are identical d, = ;, = TH = 4.
(3)
where we have used the tensor signs only to emphasize that we are dealing with expectations of infinite dimensional sets. The relations may be seen as analogous to the direct sum and direct product of the set vector SV(V) of all vector bundles. The equality of equation (3) may seem as an utterly trivial result but it is not and could never have been possible if nature had chosen for d, any other value than d, = 2 which is the only value for which the Heisenberg uncertainty relation becomes resolution independent [3-6, 9, lo]. The above result is now in complete and total agreement with two other well established results namely: (A) The analysis of the ‘3 lattice, which, although it appears to be infinite-dimensional, has an effective Hausdorff dimension dH = 4 [2]. (B) The analysis of the infinite-dimensional Cantorian spacetime which has a finite expectation value for the dimension 12 and the fractal dimension df, namely [S, 9,
101 - (n) = (dF) = 4 + 4’ = 4.23 = 4,
(4)
where 1/43 =4 + 43 and 4 is the golden mean. In addition, the topological dimension corresponding to the expectation value is exactly 4 [9, lo]. This result suggests that intersection and union are indistinguishable in four-dimensional Cantorian spacetime. Consequently we have a severe restriction on our ability to locate a Cantorian ‘point’ in space and it is even possible to say that a particle could exist in two spatially separated locations in Zcx’ at the same time. This would amount to a realistic geometrical resolution of the paradox posed by the outcome of the famous two slit experiment [9, 11, 121. CONCLUSION
We conclude that the model for a branched polymer 8 lattice [2] is an excellent analogy to visualize %‘-). Both models, although strictly infinite-dimensional, have a four-dimensional ‘appearance’ and an effective Hausdorff dimension D, = D, = dH = 4. Furthermore, the exceptional situation that the multiplication theorem and the addition theorem of independent events give the same result for the expectation value must have implications for the interpretation of the famous two slit experiment of quantum particles [9].
The Bethe lattice and the dimension of micro spacetime
1X89
Finally, it should be noted that the analogy between Cantorian-fractal spacetime and the V model open the possibility of introducing a hierarchical superstring theory where vibrating branches of a Cayley tree replace the conventional strings. In such a model there can obviously be no cut off at the Planck length, but this should not introduce great difficulties because it may be sufficient to concentrate on the effective four-dimensional core of the string-lattice space. To sum up Cantorian spacetime %(%) is not only noncommutative and nonassociative but also a random and fundamentally fuzzy geometrical manifold in which even the Euclidean norm is resolution dependent. The classical concept of a point does not exist in this geomtry. Every point, however small, has a structure which reveals itself when a sufficiently sharp resolution is employed. Only the Heisenberg uncertainity and the indistinguishability between intersection and union of the subsets of gcX) are resolution independent. Since at infinite resolution the concept points disappear completely, there should be no problems in %(-) connected to ultraviolet divergence. Finally, we note that the conclusion regarding the irreducible indistinguishability between (-17 and l,J: amounts to a realistic resoluton of the two slit experiment [12].
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9. 10. Il. 12.
Schroeder, M., Fractals, Chaos and Power Laws. Freeman, New York. 1991. Gouyet. .I. F., Physique et Structures Fractales. Masson, Paris, 1992. Ord, G. N.. Fractal spacetime: A geometric analogy of relativistic quantum mechanics. J. Phys, A. 1983, 16. 1869-1884. Ord, G. N., A stochastic model of Maxwell’s equations in 1+ 1 dimensions. lnternationaf Journal of Theoreticul Physics, 1996,35,323-326. Ord, G. N.. Conrad, M., Rossler, 0. E. and El Naschie, M. S., Nonlinear dynamics, general relativity and the quantum. Chaos, Solitons & Fractals, 1997,5,727-850. Nagasawa, M., SchrGdinger Equations and Diffusion Theory. Birkhauser, Basel, 1993. Feynman, R. P and Hibbs, A. R., Quantum Mechanics and Path Integral. McGraw-Hill, New York, 1965. Abbott, L. and Wise, A., Dimension of quantum mechanical path. American Journal offhysics. 1981,49,37-39. El Naschie. M. S. and Prigogine, I., Time symmetry breaking in classical and quantum mechanics. Chaos, Solitons & Fructals, 1996, 7,441-518. El Naschie, M. S., Rossler, 0. E. and Ord, G. N., Information and diffusion in quantum physics. Chaos, Solitons & Fractals, 1996, 7,611-819. Casti. J. L.. Paradigms Ldst. Avon Brooks, New York, 1989. El Naschie, M. S., On the uncertainty of Cantorian geometry and the two slit experiment, Chaos, So/irons & Fractab In press.
The Bethe
Lattice
and the Dimension
of Micro
Spacetime
M. S. EL NASCHIE Department of Applied Mathematics and Theoretical Physics, tiniversity of Cambridge, Silver Street. Cambridge. UK Abstract-An analogy is proposed between the Bethe lattice and the Cantorian spacetime picture of quantum physics, showing that in both cases the Hausdorff dimension of an effectively quasi-fractal meanfiled theory is equal to 4. This is identical to the upper limit of the Hausdorff dimension of a branched polymer, namely 2MI(M - I), for M = 2.0 1997 Elsevier Science Ltd
In this short note, we shall be concerned with an instructive analogy between the expectation value of the dimension of micro spacetime and a class of hierarchical lattices known as Bethe lattices $8 or simply Cayley tree [l, 21. Micro spacetime %(J-j is assumed to be an infinitedimensional transfinite Cantorian manifold [3-51. Somewhat similar to CFin dimensionality (n = m) and its geometry, the s lattice is used widely to model branched polymers and is frequently the only way to an exact solution [l, 21. Our basic idea in the present note is, in effect, to combine several of the relatively recent results obtained in the context of quantum diffusion [6] and quantum physics in fractalCantorian spacetime setting 13-51 and then show that both %‘-) and % must have the same Hausdorff dimension, namely dH = dB equal to 4. To do this, we need to recall the following results: 1. It is well-known from the work of Feynman [7] and Ord and others [3-51 that the path of a quantum particle is in general non-differentiable and fractal-like. It has been further established by Abbott et al. [8], Ord, Nottale and others that this fractal path must have a Hausdorff dimension d, = 2 irrespective of the dimension of the embedding space [9] which can be infinite. 2. There is a classical analogy between the diffusion equation and the Schrodinger equation that has been utilized by many researchers using different techniques such as analytical continuation to find a ‘realistic’ interpretation of orthodox quantum mechanics [6, 3-51. 3. It is by now quite well understood through the work of Nagasawa [6] and others, using the adjoint system method [lo], that a Schriidinger equation may be replaced by two diffusion equations in duality. In this way, Nagasawa was able to show that the Born interpretation P = (94) is more or less a derivable formula [6, lo] and an integral part of the equations of quantum mechanics. 4. Using the adjoint system method, the writer and others were able to show that the Born formula can be interpreted as a quantum probability density defined by a pair of conjugate complex Schriidinger equations which may be transformed to a pair of diffusion equations using complex time [5]. This particular interpretation is in complete harmony with (a) Eddington-Schrodinger interpretation [9], (b) Cramer’s transactional interpretation af quantum mechanical measurement [9, 111 and (c) the Wheeler-Feynman absorber theory [9-111. Combining all the results and ideas of the preceding four points, we come to the following quantum spacetime picture and the corresponding 3 lattice analogy. 1887
M. S. EL NASCHIE
1888
To develop this picture, we must first attempt to imagine a fractal path representing in effect a fractal geodesic in n-dimensional spacetime. Next, and in analogy to complex time [5] and Nagasawa’s dual diffusion [6], we imagine that we are superimposing transversally on this geodesic a second one which is supposed to be ‘running’ back in time and is thus a quasi ‘antigeodesic’, to use our earlier terminology for want of a better one. Thus any real measurement on EcX) is bound now to be taken on geodesic ‘points’ formed by the intersection or union of the two sets forming the two transversal geodesics. This will immediately imply a critical Hausdorff dimension dH equal to (Dim,r\%(“‘)=
~~=(a,=*)O(;,=2)=‘d),O’dl..
(1)
or (Dim,vS(n,)=~~=(d==2)0(;7,=2)=‘I;),0’hl,
(2)
That means that direct products and direct sums are identical d, = ;, = TH = 4.
(3)
where we have used the tensor signs only to emphasize that we are dealing with expectations of infinite dimensional sets. The relations may be seen as analogous to the direct sum and direct product of the set vector SV(V) of all vector bundles. The equality of equation (3) may seem as an utterly trivial result but it is not and could never have been possible if nature had chosen for d, any other value than d, = 2 which is the only value for which the Heisenberg uncertainty relation becomes resolution independent [3-6, 9, lo]. The above result is now in complete and total agreement with two other well established results namely: (A) The analysis of the ‘3 lattice, which, although it appears to be infinite-dimensional, has an effective Hausdorff dimension dH = 4 [2]. (B) The analysis of the infinite-dimensional Cantorian spacetime which has a finite expectation value for the dimension 12 and the fractal dimension df, namely [S, 9,
101 - (n) = (dF) = 4 + 4’ = 4.23 = 4,
(4)
where 1/43 =4 + 43 and 4 is the golden mean. In addition, the topological dimension corresponding to the expectation value is exactly 4 [9, lo]. This result suggests that intersection and union are indistinguishable in four-dimensional Cantorian spacetime. Consequently we have a severe restriction on our ability to locate a Cantorian ‘point’ in space and it is even possible to say that a particle could exist in two spatially separated locations in Zcx’ at the same time. This would amount to a realistic geometrical resolution of the paradox posed by the outcome of the famous two slit experiment [9, 11, 121. CONCLUSION
We conclude that the model for a branched polymer 8 lattice [2] is an excellent analogy to visualize %‘-). Both models, although strictly infinite-dimensional, have a four-dimensional ‘appearance’ and an effective Hausdorff dimension D, = D, = dH = 4. Furthermore, the exceptional situation that the multiplication theorem and the addition theorem of independent events give the same result for the expectation value must have implications for the interpretation of the famous two slit experiment of quantum particles [9].
The Bethe lattice and the dimension of micro spacetime
1X89
Finally, it should be noted that the analogy between Cantorian-fractal spacetime and the V model open the possibility of introducing a hierarchical superstring theory where vibrating branches of a Cayley tree replace the conventional strings. In such a model there can obviously be no cut off at the Planck length, but this should not introduce great difficulties because it may be sufficient to concentrate on the effective four-dimensional core of the string-lattice space. To sum up Cantorian spacetime %(%) is not only noncommutative and nonassociative but also a random and fundamentally fuzzy geometrical manifold in which even the Euclidean norm is resolution dependent. The classical concept of a point does not exist in this geomtry. Every point, however small, has a structure which reveals itself when a sufficiently sharp resolution is employed. Only the Heisenberg uncertainity and the indistinguishability between intersection and union of the subsets of gcX) are resolution independent. Since at infinite resolution the concept points disappear completely, there should be no problems in %(-) connected to ultraviolet divergence. Finally, we note that the conclusion regarding the irreducible indistinguishability between (-17 and l,J: amounts to a realistic resoluton of the two slit experiment [12].
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9. 10. Il. 12.
Schroeder, M., Fractals, Chaos and Power Laws. Freeman, New York. 1991. Gouyet. .I. F., Physique et Structures Fractales. Masson, Paris, 1992. Ord, G. N.. Fractal spacetime: A geometric analogy of relativistic quantum mechanics. J. Phys, A. 1983, 16. 1869-1884. Ord, G. N., A stochastic model of Maxwell’s equations in 1+ 1 dimensions. lnternationaf Journal of Theoreticul Physics, 1996,35,323-326. Ord, G. N.. Conrad, M., Rossler, 0. E. and El Naschie, M. S., Nonlinear dynamics, general relativity and the quantum. Chaos, Solitons & Fractals, 1997,5,727-850. Nagasawa, M., SchrGdinger Equations and Diffusion Theory. Birkhauser, Basel, 1993. Feynman, R. P and Hibbs, A. R., Quantum Mechanics and Path Integral. McGraw-Hill, New York, 1965. Abbott, L. and Wise, A., Dimension of quantum mechanical path. American Journal offhysics. 1981,49,37-39. El Naschie. M. S. and Prigogine, I., Time symmetry breaking in classical and quantum mechanics. Chaos, Solitons & Fructals, 1996, 7,441-518. El Naschie, M. S., Rossler, 0. E. and Ord, G. N., Information and diffusion in quantum physics. Chaos, Solitons & Fractals, 1996, 7,611-819. Casti. J. L.. Paradigms Ldst. Avon Brooks, New York, 1989. El Naschie, M. S., On the uncertainty of Cantorian geometry and the two slit experiment, Chaos, So/irons & Fractab In press.