Making a Wavefunctional representation of physical states congruent with the false vacuum hypothesis of Siney Coleman A. W. Beckwith Department of Physics an Texas Center for Superconuctivity an Avance Materials at the University of Houston Houston, Texas 774-55 USA ABSTRACT We examine quantum ecay of the false vacuum in the riven sine-goron system an show that it is consistent with an matches up with the soliton-antisoliton (S-S ) separation istance obtaine from the Bogomol nyi inequality. This inequality permits construction of a Gaussian wave functional representation of S-S nucleate states an is consistent with the false vacuum hypothesis. Corresponence: A. W. Beckwith: projectbeckwith@yahoo.com PACS numbers: 3.75.Lm,.7., 7.45.Lr, 75.3.Fv, 85.5.Cp
INTRODUCTION In this paper, we apply the vanishing of a topological charge Q to show how the Bogomol nyi inequality 6 can be use to simplify a Lagrangian potential energy term so that the potential energy is proportional to a quaratic scalar fiel contribution. In oing so, we work with a fiel theory featuring a Lorenz scalar singlet value fiel in D imensional spacetime. Topological charges an inequalities exist an hol respectively for D imensional theories featuring scalar an singlet value fiels; only for D =. For D >, the Lorentz scalar fiels must be D-plets! We use the D = imensional case for escribing the ynamics of quasi-one-imensional metallic materials in our conense matter example. We then escribe how the quantum ecay of a false vacuum, contributes to our problem. For forming the wavefunctionals in our new functional integration presentation of transport theory, we employe a least action principle that Siney Coleman use for WKB-style moeling of tunneling. As a sign of its broa scientific interest, for over two ecaes several quantum tunneling approaches have been propose to this issue of the quantum ecay of the false vacuum. One is to use functional integrals to compute the ucliean action ( bounce ) in imaginary time. This permits inverting the potential an moifying what was previously a potential barrier separating the false an true vacuums into a potential well in ucliean space an imaginary time. The ecay of the false vacuum is a potent paraigm for escribing ecay of a metastable state to one of lower potential energy. In conense matter, this ecay of the false vacuum metho has been use 3 to escribe
nucleation of cigar-shape regions of true vacuum with soliton-like omain walls at the bounaries in a charge ensity wave. We use the ucliian action so that we may invert the potential in orer to use WKB semiclassical proceures for solving our problem. Another approach, 4 using the Schwinger proper time metho, has been applie by others 5 to calculate the rates of particle-antiparticle pair creation in an electric fiel for the purpose of simplifying transport problems. Our metho simplifies what were excessively complicate solutions an fits well with more abstractly presente treatments of transport theory. 6 We also mathematically elaborate upon the S-S omain wall paraigm 7,8 so that a tunneling Hamiltonian formalism we present elsewhere has the kinetic energy information, whereas the wavefunctionals erive here contain the tilte potential contribution the false vacuum hypothesis gives us for physics problems. BASIC TCHNIQUS USD IN THIS PAPR In this stuy, we apply the omain wall physics of S-S pairs to obtain a quaratic scalar value potential for transport physics problems involving weakly couple scalar fiels. We foun it necessary to write up the energy of a soliton kink in the beginning an then to apply the Bogomol nyi inequality to obtain a greatest lower boun to an energy of the kink expression integrate over spatially (to obtain a mass of this kink), which is a topological charge. After this energy/mass representation of the soliton kink is moifie by the Bogomol nyi inequality, we can use the boun on our moifie potential to simplify a ucliian least action integral For a S-S pair, the topological charge Q (so esignate ue to omain walls) vanishes. If we use ucliian imaginary time, the least action integral of our wave functional will be change from q. (a) below to q. (b) by using time i ( time). 3
( ) ( ) ( ) V x i D / exp h (a) transforms to ( ) ( ) ( ) V x D / exp h (b) We shoul note that q. (b) has an energy expression of the form () ( ) ( ) ε V x (a) q. (a) has a potential term that we can write as () ( ) ( )... 4 T H O C C V (b) Furthermore, even after we invert our potentials, we can simplify our expression for the potential by proceures that eliminate the scalar potential terms higher than by consiering the energy per unit length of a soliton kink. This is given by A. Zee, after rescaling to ifferent constants, as () ( 4 ~ ϕ λ ε ) = x x (3) with a mass of the kink or antikink of this given by () x x M ε ~ (3a) to be boune below, namely, by use of the Bogomol nyi inequality ( ) Q x x M λ µ µ ϕ λ 3 4 (4) 4
where Q is a topological charge of the omain wall problem. We efine conitions for forming a wave functional via the Bogomol nyi inequality an the vanishing of the topological charge Q, as given by q. (5):,9 Ψ = > c [ ] ) ( D ) exp( α x (5) C We presuppose, when we obtain q. (5), a power series expanasion of the ucliian Lagrangian, L about C. The first term of this expansion, r L = = ( ) = ε() (6) O is a comparatively small quantity that we may ignore most of the time. Furthermore, we simplify working with the least action integral by assuming an almost instantaneous nucleation of the S-S pair. We may then write, starting with a Lagrangian ensity ζ, τ x ζ t x ζ value t x L (7) P P Quantity t P in equation 7 is scale to unity. q. (7) allows us to write our wave functional as a one-imensional integran. We calle the t as a unit interval of time in this P calculation. q. (7) nees consierable explanation. To o this, break up the Lagrangian ensity as with ( ) V ( ) ζ (8) ( ) = ( τ ) ( ) r (9) 5
where the ucliian imaginary time is over such a short interval that we, instea, look at the spatial variation accoring to setting the time varying contribution of the phase as a uniform constant term, so we look at r ( ) ( ) (9a) an then look at the integran as with τ x ζ = t ε ( () (9b) P ( ε () x ( ) V ( ) () Then we have to look at the behavior of ( ) δ ( x L / ) δ ( x L / () n n ) which woul represent the behavior of test functions converging to Dirac elta functions as n Furthermore, we shoul look at the behavior of, if N is very large x x ( ) x [ δ ( x L / ) δ ( x L / ) ] [ δ ( x L / ) ] x [ δ ( x L / ) ] N [ / ] < n N N N N () where I am, for this example moeling via use of the Gaussian ( normal) probability function, which tens to a quite large elta function for all N N ( ) ( ~ ) ( ) ~ x ± L / δ x N / π exp( x N / 4) δ (3) where for all N values we have N 6
δ ~ ~ = (4) N ( x ) x So, then, we are analyzing this problem accoring to a finite contribution of x n N < with contributions about the omain walls of ( ) x ±L / assuming a thin wall approximation, as illustrate by Fig.. [Insert Fig. here] Introucing omain wall physics via q. (6) an q. (7) allows us to use a least action integral interpretation of WKB tunneling as the starting point to our analysis. This permits us to write our wave functional as proportional to,9 ψ c ~ ( β L τ ) exp (5) with the Lagrangian treate as V 3 L L = ( ) O = ( ) 3 = 4! V 4 4 ( ) 4 = 3! 3 V (6) We shoul be aware that for a wick rotation, when t = i τ that for imensions x = i x with x = τ x, an then we will set =, effectively leaving us with use of ~ ϕ λ ε () x = ( ) for a soliton kink. We also use a x 4 conserve current quantity of, µ µν J = ε ν (7) ϕ with a topological charge of 7
Q x J () x = [ ( ) ( ) ] (8) ϕ Note that the enominator ϕ is not the same as () x! In Zee, the ϕ term is ue to his setting of two minimum positions for for a ouble well potential. We fin that if we have meson type behavior for the fiel () x, this charge will vanish. It is useful to note that if we look at the mass of a kink via a scaling µ λ with M efine as the same as the energy of a soliton kink given in q. (3), with a subsequent mass given in q. (3a), that we have, via using q. (4). So that a b a b, an inequality of the form given by M Q (9) with mass M in terms of units of 4 ( ) = ( ) ( ) 4 4 µ µ. If we note that we have 3 λ in one imension, we physically use our topological current as a vanishing quantity from the kinetic term an the fourth orer term both in a current as a vanishing quantity from the kinetic term an as an expansion of the potential about. Then we can write L Q ( C ) { } () where Q () Due to a topological current argument (S-S pairs usually being of opposite charge) an 8
{ } { } { } α where if we pick : A B gap () ({ } { } { } ) A B gap V ( ) V ( ) F T (3) This means a wavefunctional with information from a inverte potential as part of a transport problem of weakly couple systems along the lines suggeste by Tekeman. We foun our weakly couple systems eliminate the cross terms in our erivation of a functional integral an for D =, can write more generally the initial configuration of the form Ψ exp{, i [ () x ] = c α x[ () x i c i ] } (4a) Ci which is i { } [ () x ] = c exp α' x L () Ψ x = c exp{ α' S }. (4b) in aition, we woul also have a final state immeiately after tunneling,, Ψ exp{ x, f [ Cf f Cf ] } (4c) [ () x ] = c x β() x () x () In the case of a riven sine-goron potential system, the initial state is similar to Coleman s false vacuum bounce representation. The final state can be approximate as a 9
moifie Gaussian centere about a final fiel configuration of () x that inclues a bubble in which has tunnele through the barrier into the true vacuum state, creating one or more soliton omain walls at the bounary between true an false vacuums () x insie the tunnel barrier. Furthermore, we have that Cf V =, ε F. ε ε (5) that is then tie in with the Bogomol nyi inequality formulation of q. () where the topological charge Q ε. We also have in the case of a riven sine-goron potential a situation where we can generalize our wave functionals as, Ψ c i { [ c ] } i [ () x ] = c exp α x () x Ci exp α i ~ [ ] Ψ, x F initial (6a) an Ψ c f [ () x ] = c exp x α () x () x Cf exp α f ~ [ ] Ψ, x T final C f (6b) where a riven sine-goron system is of the form 9 (assuming C a >> C b )
V () ( cos) ( C a C b ) (7a) C i, f T π (7b) We also assume that c i, with the i being either or, will take into consieration the contributions enote from q. (). Furthermore, where x is the initial an final state equilibrium configuration of phase, the wavefunctionals so obtaine permit us to write wavefunctionals that obey the extremal conition of,9 ci,cf () δ δ () ( L ) i, f τ x Ci, Cf (8) which is a further tie in with Siney Coleman s fate of the false vacuum hypothesis. CONCLUSION It is straightforwar to construct wavefunctionals that represent creation of a particular event within an embeing space. Diaz an Lemos 3 use this technique as an example of the exponential of a ucliian action to show how black holes nucleate from nothing. This was one in the context of e Sitter space; Diaz an Lemos 3 use a similar calculation with respect to nucleating a e Sitter space from nothing. The ratio of the moulus of these two wavefunctionals is use to calculate the probability of Black hole nucleation within a e Sitter space, which is the general embeing space of the universe. This trick was also use by Kazumi Maki 4 to observe a fiel theoretic integration of conensates of S-S pairs in the context of bounary energy of a two-imensional bubble of space-time. This two-imensional bubble action value was minus a contribution to the action ue to volume energy of the same two-imensional bubble of space-time. Maki s 4
probability expression for S-S pair prouction is not materially ifferent from what Diaz an Lemos 3 use for black hole nucleation. What we have one is to generalize this technique to constructing wavefunctional representations of false an true vacuum states in a manner that allows for transport problems to be written in terms of kinetic ynamics as they are given by a functional generalization of a tunneling Hamiltonian. It also allows us to isolate soliton/instanton information in a potential fiel that overlaps with a Gaussian wavefunctional presentation of soliton/instanton ynamics. We believe that this approach will prove especially fruitful when we analyze nucleation of instanton 5 states that contribute to lower imensional analysis of the configurations of known physical systems (e.g., NbSe 3 ).,9 This approach to wavefunctionals materially contributes to calculations we have performe with respect to I- curves fitting experimental ata quite exactly,9 an in a manner not seen in more traitional renerings of transport problems in conense matter systems with many weakly couple fiels interacting with each other. 6
RFRNCS 6 A. Beckwith; arxiv math-ph/4653; A. Zee, Quantum fiel theory in a nutshell, Princeton University Press 3, pp.6-63 an pp.79-8. S. Coleman; Phys.Rev.D 5, 99 (977). 3 I.V. Krive an A.S. Rozhavskii; Soviet Physics JTP 69,55 (989). 4 J. Schwinger, Phys.Rev.8, 664 (95). 5 Y. Kluger, J.M. isenberg, B. Sventitsky, F. Cooper an. Mottola; Phys.Rev.Lett. 67,47 (99). 6 For a review, see R. Jackiw s article in Fiel Theory an Particle Physics, O. boli, M. Gomes, an A. Samtoro, s. (Worl Scientific, Singapore, 99); Also see F. Cooper an. Mottola, Phys. Rev. D36, 34 (987); S.-Y. Pi an M. Samiullah, Phys. Rev. D36, 38 (987); R. Floreanini an R. Jackiw, Phys. Rev. D37, 6 (988); D. Minic an V. P. Nair, Int. J. Mo. Phys. A, 749 (996). 7 J. H. Miller,Jr., C. Oronez, an. Proan, Phys. Rev. Lett 84, 555(). 8 J. H. Miller, Jr., G. Carenas, A. Garcia-Perez, W. More, an A. W. Beckwith, J. Phys. A: Math. Gen. 36, 99 (3). 9 A.W. Beckwith, Classical an quantum moels of ensity wave transport: A comparative stuy. PhD Dissertation,. Davison Soper, Classical Fiel theory, Wiley, 976, pp -8.,eqn 9.3. 3
S. Ciraci,. Tekman ; Phys.Rev. B 4, 969 (989). Hermann G. Kümmel, Phys. Rev. B 58, 6 65 (998). 3 O. Dias, J. Lemos: arxiv :hep-th/368 v 7 Oct 3. 4 K. Maki; Phys.Rev.Lett. 39, 46 (977), K. Maki Phys.Rev B 8, 64 (978). 5 Javir Casahoran, Comm. Math. Sci, Vol, No., pp 45-68. 6 W. Su, J. Schrieffer, an J. Heeger, Phsy Rev Lett. 4, 698(979). 4
FIGUR CAPTION Fig. : volution from an initial state i [ ] to a final state f [ ] for a ouble-well potential (inset) in a -D moel, showing a kink-antikink pair bouning the nucleate bubble of true vacuum. The shaing illustrates quantum fluctuations about the initial an final optimum configurations of the fiel, while (x) represents an intermeiate fiel configuration insie the tunnel barrier. The upper right han sie of this figure is how the fate of the false vacuum hypothesis gives a ifference in energy between false an true potential vacuum values which we tie in with the results of the Bogomol nyi inequality. 5
Figure Beckwith 6