The Scalar Particles in Casimir Vacuum State of the Relativistic Quantum Field System in Living Cells

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Bulgarian Journal of Agricultural Science, 18 (No 6) 2012, 819-826 Agricultural Academy

The scalar particles in Casimir vacuum state of the relativistic quantum field system in living cells

Ts. D. Tsvetkov1 and G. Petrov2 1 Institute for Cryobiology and Food Technologies, BG - 1111 Sofia, Bulgaria 2 Institute for Nuclear Researches and Nuclear Energy, BG – 1113 Sofia, Bulgaria Abstract

Tsvetkov, Ts. D. and G. Petrov, 2012. The scalar particles in Casimir vacuum state of the relativistic quantum field system in living cells. Bulg. J. Agric. Sci., 18: 819-826

The mathematical description of the Casimir world is based on the fine play between the continuity and the discrete. The discrete is more remarkable than the continuity things in this case as any one wave quantum field Casimir vacuum state defined on the involutes algebraic entities so that the quantum wave functional, i.e. the ground state, is non positive as remembers of the Hilbert space with indefinite metric and can be consider as a solution of the Schrödinger functional equation with impulse effect. The discrete things can be singularities, bifurcations and destroyed scaling behavior of the relativistic scalar quantum field system and his Casimir ground state in living cells as global properties of the quantum vacuum state interacting with external classical field modelled in our case by boundary conditions. The classical field theory is a theory of the continuity also describes the principle of the shot-range interaction in the nature. The twenty century has given the quantum field theory that is the theory of the discrete world of the quantum entities. The global effects from the quantum field theory by the interactions of the elementary particle are the studying of the phenomenon of structure of the elementary particle hadrons by deep inelastic scattering and a vacuums structure of every one quantum field system, e g. the relativistic quantum field and his vacuum state with a given boundary conditions. The question above the possibility to find the complicate phenomenon connected with the existence of the life and the living systems his place in the mathematical frame by the intersection between the classical and the quantum describing of the world of any one concrete quantum field theory by the contemporary ground state of the theoretical biological and nanophysics problems is open by the consideration of high topographical complementarities by the London- and Casimir forces involved importantly in the highly specific and strong but purely classical physical thermodynamically and quantum physi- cally complexity of elementary living cells by enzymes with substrates, of antigens with antibodies, etc. The theory of the “super transportation” based on the “Einstein condensation” of the bosons particles are given strong mathematical by N.N. Bogolubov in 1946 and has been defined a quasiparticles as excitations in a so-called collective state as a superposition between an excited particles state and the ground state of the quantum system. By the consideration of the living cells, it is necessary to consider the quantum field concepts by Casimir vacuum state in observation scalars (here called as Pterophillum scalarе) of the sea relativistic quantum scalar field system. From the new results by the contributions of the environmental freezing-drying and vacuum sublimation (Zwetkow 1985; Tsvetkov, Belоus 1986, Tsvetkov et al. 1989, Tsvetkov et al. 2004 – 2012) is hopped that by the significant form expressed e.g. by the scaling behaviours of the invariant entities by the energy impulse tensor described the elementary living cells and systems will be possibly to describe the biological expressions at the standpoint of the nanophysics by means the behaviour of the concrete relativistic quantum field system, e.g. sea virtual quantum scalars in the physical Casimir vacuum with a boundary conditions on every one surface S too.

E-mail: [email protected] 820 Ts. D. Tsvetkov and G. Petrov

At the molecular level (Mitter and Robaschik, 1999) the thermodynamics behaviour is considered by quantum electromagnetic field system with additional boundaries as by the Casimir effect between the two parallel, perfectly conducting square plates (side L, distance d, L > d), embedded in a large cube (side L) with one of the plates at face. For further considerations we can suppose that the demon of Maxwell slide the moved plate with a constant velocity v and counted the seeing scalars for j = 0, …, 2n, where j is the number of the scalars with the minimum energy equal to the energy µ of the “zero fluctuations” levels by the hyperbolical rotation and reflection of the fixed 4-point y0 from Minkowski space between the plates from the demon of Maxwell counted.

Key words: Casimir effect, living cells, lyophilization, nanophysics, singularities, scaling principle, Casimir vacuum state

Introduction To day, it is clear that hadrons are no more so elementary particles. By consideration of the non-forward The mathematical description of the Casimir world Compton Effect in deep inelastic scattering, they have is based on the fine play between the continuity and a structure (Petrov, 1978) as the atom has understood discrete. The discrete is more remarkable than the con- it. In the beginning of the twenty century was clear that tinuity things as any one wave quantum field Casimir the atom is no more not devisable as it has been toughed vacuum state defined on the involutes algebraic field by the ancients Greeks and it is a particle with a struc- operator entities so that in this case the quantum wave ture. So we have to consider an important scaling prin- functional is non positive as remembers of the Hilbert ciple as the scaling destroyed in longitudinal and not space with indefinite metric and is a solution of the destroyed in the squared direction by the describing of Schrödinger functional equation with impulse effect the Casimir world based of this fine play between the too. The discrete things can be singularities, bifurca- continuity and the discrete where we have to consider tions and distortions of the scaling behaviour of the the singularities, the bifurcations and the restructur- Casimir ground state of the relativistic scalar quantum ings of the vacuum state of every one quantum field field system in living cells as global properties of the system defined on the space-time of our world inter- vacuum state interacting with external classical field acting with a given boundary conditions on a generic modelled in our case by boundary conditions. The surface S. It is clear also that the quantum vacuum has classical field theory is a theory of the continuity also a structure and is no more vacuum also a void. describes the principle of the shot-range interaction in From the calculating of Casimir effect is clear that the nature. The twenty century has given the quantum the Casimir force is from significant importance for the field theory that is the theory of the discrete world of nanophysics and so also for life and living systems. If the quantum field entities. The global effects from the the scalars are the particles transported the interaction quantum field theory by the interactions of the elemen- between the quantum ground state and the plates by tary particle are the phenomenon of the structure of the Casimir effect in the sense of the short-range inter- the hadrons and a quantum vacuums structure of every action than the Einstein condensation can be observed one quantum field system defined in Hilbert space with for the zero impulse of the particles with a scaling be- indefinite metric, e g. the relativistic quantum field and haviour in the squared direction towards the moving his vacuum state with a given boundary conditions. demon of Maxwell so that in the longitudinal direction The axiomatic algebraic quantum field methods by the the impulse is not zero but less then the impulse of the description of the quantum particle physics world have kinetic energy of the zero fluctuation of the vacuum important remarkable successes by the understanding and the scaling is destroyed by this scalars impulse that of the structure of the elementary particle and the vac- is equal of the impulse of the “energy of the zero fluc- uum state. tuations” of the vacuum. So the virtual quantum par- The Scalar Particles in Casimir Vacuum State of the Relativistic Quantum Field System in Living Cells 821 ticles and the zero fluctuations of the vacuum take one as a light like Minkowski space vector and meaning. yµ and yµ are fixed 4-vectors of the Minkowski 2(n-½j) -2(n-½j) space-time. Main result Further, we define by the following relations κxµ = τ rxµ̃ + (x + τ rx)̃ µ f with At the first, we take the reflecting and the hyperboli- j ½ j κ f = y -2 (x τ rx)((1̃ + 4κ2x2y 2(x τ rx)̃ -2)½ - 1) cal turns on the mirror points in the Minkowski space κ 0 ½ j 0 j µ µ µ µ y , y , y y 2 r 2 2(n - (j-1)) -2(n - j) 2(n - j) , -2(n - (j+1)) and the point τ ̃ ∈ ½ ½ ½ ½ for y2(n - ½j) = ¼ (x + j x) , t (t-2(n - ½j), t2(n - ½j) ], µ  µ  n = 0,1,2, ..., j = 0,1,2, ..,2n, x = (ct,x) and y0 = (ct0,y) between the plates so that ∈ 3 µ l µ̃ l ̃ µ t (t-2(n - j), t2(n - j) ], n = 0,1,2, ..., j = 0,1,2, ..,2n for x κ’x = τ x + (x + τ x) fκ with ½ ½ j ½ j ’ 3 ∈ (0, d ], or x ∈ (d , L), f = y -2 (x τ lx)((1̃ + 4κ’2x2y 2(x τ lx)̃ -2)½ - 1) 0 0 κ’ 0 ½ j 0 j where µ = 0,1,2,3, and n is the reflecting number of µ for y 2 = (x + τ lx)̃ 2, t ∈ (t , t ], the seeing from the demon of Maxuel point y from -2(n - ½j) ¼ j -2(n - ½ j) 2(n - j) 0 ½ the Minkowski space-time at the time t = t0 between n = 0,1,2, ..., j = 0,1,2, ..,2n, the not moved and the moved plate with the constant κxµ = τ rxµ̃ + (x + τ rx)̃ µ f with velocity v so that 2n-1 ½ 2n-1 κ 2 2 2 2 2 y = y = y = y = y = f = y -2 (x τ rx)((1+4̃ κ2x2y 2(x τ rx)̃ -2)½ - 1) 2(n - ½(j-1)) -2(n - ½j) 2(n - ½j) 2(n - ½(j+1)) 0 κ 0 ½ 2n-1 0 j 2 2 2 2 for y 2= (x + τ rx)̃ 2, t ∈ (t , t ], so that the follow- (ct )2 - y = (ct ) - y - y 3 ≠ 0, 1 ¼ 2n-1 -1 1 0 0 ⊥ 0 ing bounded open domains of double cons are defined _ 3 3 y ∈ (0, d ], y ∈ [d , L) l l + 0 0 0 0 D = D l ̃ = V l ̃ ∩ V with the basis S l ̃ and κx, τj x τj x κx κx, τj x 3 3 µ l µ µ t-2(n - j) = t2(n - (j+1)), y-2(n - j) = -y2(n - (j+1)) , t2(n - (j-1)) = κ τ ̃ ½ ½ ½ ½ ½ the axis [ x , j x ] = y2(n - ½j) fκ, t , y 3 = - y 3. Also we have a so called _ -2(n - j) 2(n - (j-1)) -2(n - j) r r + ½ ½ ½ D = D r ̃ = V r ̃ ∩ V with the basis S r ̃ µ µ κ’x, τj x τj x κ’x κ’x, τj x lexicographic order for y and y µ r µ µ 2(n - ½(j-1)) -2(n - ½j) ; and the axis [κ’x , τ x̃ ] = y f . µ µ  j -2(n - ½j) κ’ y and y , and we change not the y⊥ 2(n - ½j) -2(n - ½(j+1)) Further we can define for the impulse Minkowski by the reflections and the hyperbolically rotations. Fur- space by means of the following relation and fixed im- ther we can defined by the distinguishing marks “l” = pulse four vector kµ = (0, 0, 0, k3) as by the P.N. Bogol- left and “r” = right the following relations between the ubov et.al. and G. Petrov 1978 point x between the plates and the reflecting see points µ r µ̃ µ r µ µ µ -2 2 qκ = q + k fκ with x̃ = x + (y2(n- j) y0 )((xy2(n- j)) - ½ ½ -2 r -2 2 2 r -2 µ f = k (k q)((1̃ + κ q k (k q)̃ )½ - 1), x2y 2)½ - y (xy )y -2 = κ 0 2(n-½j) 2(n-½j) 0 xµ + y µ ((xy )y -2)((1 - x2y 2( xy )-2)½ - 1) = q µ = lqµ̃ + kµ f with 2(n-½j) 2(n-½j) 0 0 2(n-½j) κ’ κ’ µ µ x + y f, for t ∈ (t , t ] and f = k-2 (k lq)((1̃ + κ’-2q2k2(k lq)̃ -2)½ - 1) and -2(n-½j) -2(n - ½j) 2(n - ½j) κ’ f = ((xy )y -2)((1 - x2y 2( xy )-2)½ - 1) 2(n-½j) 0 0 2(n-½j) qµ = (q µ + q µ), kµ = (q µ - q µ), lq2̃ = rq2̃ = 0, q 2 ½ κ κ‘ ½ κ κ‘ κ and κ-2 2 2 κ -2 2 2 32 = q , qκ’ = ’ q , k = - k l µ µ µ -2 2 x̃ = x + (y-2(n- j) y0 )((xy-2(n- j)) - ½ ½ 2 2 2 2 2 02 12 22 32 2 µ κ κ x2y 2)½ - y (xy )y -2 = ( qκ + qκ )/2 = q = q – q - q - k = m , where 0 -2(n-½j) -2(n-½j) 0 m is the scalars mass at the rest and k3 = q3 for destroyed xµ + yµ ((xy )y -2)((1 - x2y 2( xy )-2)½ - 1) = -2(n-½j) -2(n-½j) 0 0 -2(n-½j) scaling behaviour in the longitudinal direction. xµ + yµ f’, f = ((xy )y -2)((1 - x2y 2( xy )-2)½ - -2(n-½j) -2(n-½j) 0 0 -2(n-½j) It is assumed the local quantum scalar wave field -1), for t ∈ (t , t ], n = 0,1,2, ..., j = 0,1,2, ..,2n, system under consideration have a boundary generic -2(n - ½j) 2(n - ½j) 822 Ts. D. Tsvetkov and G. Petrov surface S for his ground state or the so called Casimir the fixed one – the free vacuum region with j = 2n-1 vacuum state, fixed or moved with a constant veloc- scalars, described by the impulse Schrödinger wave ity v parallel by demon of Maxwell sliding towards the functional fixed one boundary, which do surgery, bifurcate and Ψ (φ, t) = Ψ(f , t), by t ∈ (t , t ], n = 0, 1, 2, separate the singularity points in the manifold of the ακ κ -2(n - ½j) 2(n - ½j) non local virtual scalars of the quantum field system ….; j = 0,1,2,…,2n; from some others vacuum state as by Casimir effect t = t , t = t for j = 2n-1, t = t , -2(n - ½j) 0 2(n - ½j) 1 -2(n - ½ (j-1)) -1 of the quantum vacuum states for the relativistic quan- t = t for j = 2n tum fields, and which has the property that any virtual 2(n - ½j) 0 quantum particle which is once on the generic surface and by given fulfilled operators equation in the case of S remains on it and fulfilled every one boundary condi- the canonical Hamiltonian local quantum scalar field tions on this scalar quantum system with a non positive system in a statement of equally times. Casimir vacuum state as remember of the Hilbert space 0 0 2 2 2 3 2 ½ x = ct = κx = ct2(n - (j - κ)) = (κ x + y + (y 2(n - (j- κ)) ) with infinite metric defined on the involutes algebra of and ½ ┴ ½ the field operators A(f) for the solution f of the Klein- 0 0 x = ct → κ’x = ct-2(n - j - κ’)). = Gordon wave equation given by covariant statement ½( (κ 2x2 + y 2+ (y3 )2)½ -2(n - ½(j- κ’) □ f(xµ) = (∂2 – (∆ + m2))f(xµ) = J(xµ), (1) ┴ ct with 0≤κ’<τ<κ≤1, x → y (2) ┴ ┴ 2 2 2 By the definition the canonical non local field φ(κx) where □ is a d’Lembertian and ∆ = ∂ x + ∂ y + ∂ z is a Laplacian differential operators and xµ∈ M4 µ = and impulse π(κx), κxµ ∈ M4, µ = 0, 1, 2, 3, correspond- 0, 1, 2, 3, (x, y, z, ct) = (xl, x0) with l = 1, 2, 3 is a point ing to implicit operator valued covariant field tempered 4 ∈ 4 of Minkowski space M and the J is a current of the functional A(fκ), where fκ SR(R ) is a test function sea virtual scalars of the sea relativistic quantum scalar of this reel Swartz space, fulfilled all axioms of Whit- fields system. man and acting in the functional Hilbert space Ĥ with Also the ground states of the local quantum scalar infinite metric and a Fok’s space’s construction, e.g. a fields system defined in the Minkowski space-time ful- direct sum of symmetries tensor grade of one quantum filled every one boundary conditions interact with the field’s particle space 1Ĥ : boundary surface S by the help of the non local virtual ∞ ×n scalars and so the Casimir vacuum state has a global- Ĥ = ĤF(Ĥ1) = × sym Ĥ1 ly features, e.g. the energy of the “zero fluctuations”1 n=0 and hens the so called Casimir effect with calculable can be given for physical representation of the scalar Casimir force inversely proportional to the 4-th grade quantum field system by the formulas, of the distance between the plates. φ(ακ) = A(fκ), (3) Examples of such boundary surfaces S of impor- π(α ) = A(∂ f ), (4) tance for the living cells are those in which is the sur- κ ct κ face of a fixed mirror at the initial time t = 0 in con- for α = α (x1, x2, y3 , ct ), with y3 = κ κ 2(n - ½(j-κ)) 2(n - ½(j-κ)) 2(n - ½(j-κ)) tact with the local quantum scalar fields system in his κx3 and y 0 = κx0 by fulfilled Klein-Gordon equa- 2(n - ½(j-κ)) simple connected vacuum region – the bottom of the tion in a covariant statement for massive and massless sea of the virtual non local massless scalar quantum scalar fields field particles, for example given by the 4-impulse q̃ τ K A(f ) = A(K f ) = A(J) and the generic free surface of the parallel sliding mir- m κ m κ ror moved, e.g. by demon of Maxwell at the time t0 and KA(f) = A(Kf) = 0. (5) with a constant velocity v towards the fixed one or the It is knowing that matrix elements of the current <φ| 2 free vacuum surface of the local quantum scalar wave J(qτ ̃) |0>, i.e. for j = 0 have a singularities at q̃τ = 0 which field in contact with the mirror parallel moved towards can be interpreted as a presence in Hilbert space Ĥ of 1 Actually, this property is a consequence of the basic assumption by local quantum wave field theory that the wave front of the local quantum wave field system by his ground state propagate on the light hyperplane in any contact space (called “dispersions relations” too) and can be described mathematically as a non local virtual topological deformation or zero fluctuation which depends continuously on the time t. The Scalar Particles in Casimir Vacuum State of the Relativistic Quantum Field System in Living Cells 823 the massless scalars Goldstones bosons and the appear- The obviously necessity to take in consideration the ance of the action at the distance. But this resemblance quantum field concepts by observation macroscopi- is only formal and by going to the physical representa- cally objects present from infinity significant number tion the scalar Goldstones bosons disappears. That is of particles and to be found by low temperatures is one of the indications of the Higgs-mechanisms, e.g. following from the elementary idea. Consider e.g. the effect of the mass preservation from the vector fields Casimir vacuum present from n level of the energy of by spontaneous destruction of the gauge group (or the “zero fluctuations” and occupied by j = 2n-1 virtual scalar Goldstones bosons are “swallowing up”) and so scalar bosons to be found in a volume V. In so one vac- it is to observe, that the Casimir force is to be observed uum state every virtual scalars is surrounded closely only by quantum electromagnetic field system with a from the neighbouring particles so that on his kind get massless real photon and asymmetrical Casimir vacu- a volume at every vacuum energy level of the order V/n um state where the scaling behaviour by the fermions ~ ((|y |2 + (y3 )2)½)3 Also every virtual scalar at the ⊥ 2(n - ½j) fields is destroyed. state with the smallest energy possess sufficient energy For simplicity here, we have considered a domain of the “zero fluctuations” given elementary by of space-time containing any one massless scalar field 2 m = E ~ (2m)-1 ((|q |2 + k3 )½)3~ (2m)-1((|y |2 + ϕ(x) defined at the point of the Minkowski space-time 0 ⊥ ⊥ (y 3)2)½)-3 ~ (2m )-1(n/V ) ⅔, at the fixed time t. Further a concrete massless field 2(n - ½j)  ϕ(t, y) is considered as a Banach valued vector state for k3 = q3 and |q |2 → 0 and for obeying the impulse wave equation in a Banach space, ⊥ 3  2  2 3 ⊂ n → and | y⊥| → the lim | y⊥| /2y = defined over the space Ωt M at the time t = t0 and ∞ ∞ 2(n - ½j) 3 3 ∈ y = y 0 in the Minkowski space-time M. By impos- = const [0, 1], (6) ing suitable boundary conditions for any one quantum In addition, the distance between the ground state field system considered as any one relativistic quantum and the first excited level of the single see massless sca- fieldϕ (x) fulfilled the Klein-Gordon equation, the total lar will be of the some order that is for E0 too. It follows fields energy in any domain at the point (ct,x) from the that if the temperature of the vacuum state of the rela- Minkowski space-time can be written as a sum of the tivistic sea quantum field system is less then the some zero mode energy for one critical temperature Tc of the order of the tempera- t ∈ (t , t ], n = 0,1,2, ..., j = 0,1,2, ..,2n and -2(n - ½j) 2(n - ½j) tures of the Einstein condensation, than in the Casi- 3 ∈ 3 ∈ x (0, d0], x [d0, L), mir vacuum state there are not the excited one-particle  ≤ n = 0,1,2, ..., j = 0,1,2, ..,2n so that | x⊥| . R = states. 2 3 2 ((ct) - (x ) )½ By the calculation of the energy of the “zero fluctu- is fulfilled and the ground state of this concrete ations” of the Casimir vacuum state the Green function quantum field system must be conformed by this suit- as solution of massless Klein-Gordon equation with the able boundaries and so we can modelled the interaction δ-function as a source play an important role. of the concrete quantum field system to the external (∂ 2 – ∆)D(xµ) = δ(xµ) classical field by means of this suitable boundaries. ct Our interest is concerned to the vacuum and espe- ∞ and D(·) = ∑ (D(x - y2n) + D(x - y-2n) - cially the physical Casimir vacuum conformed by the n = 0 suitable boundary conditions. - D(x - y2n-1)) - D(x – y-2n-1)))) (7) Nevertheless, the idea - that the vacuum is like a ground state of any one concrete relativistic quantum ∫ 4 ̃ ̃ 2̃ ε where D(x - y2n) = d q exp [-iq(x - y2n )/( q - i ) = field system - is enormously fruitful for the biological - i(2π)-2 (x - y )-2 . systems in the nanophysics and cellular cryobiology 2n too, i.e. it is to consider the time arrow in the systems Therefore, the temperature for Casimir effect is not with a feedback. from significances. 824 Ts. D. Tsvetkov and G. Petrov

The Higgs Mechanism <φ> = µe iϑ0 → µe iϑ0 + iα(x) (10) and therefore the symmetry is spontaneously bro- Gauge symmetry seems to prevent a vector field ken. In addition, it is the case where the ground state from having a mass. of the relativistic quantum field system is asymmetri- This is obvious once that can be realized a term cal and the scaling behaviour of the scalar field is de- in the Lagrangian like m2A Aµ is incompatible with µ stroyed. Let it be now the theory around one of these gauge invariance. vacua, for example <φ> = µ, can be studied by writing However, certain physical situations seem to re- the field φ in terms of the excitations around this par- quire massive vector fields. This happened for ex- ticular vacuum ample during the 1960s in the study of weak interac- iϑ(x) tions. The Glashow model gave a common description φ(x) =[µ + (1/√2)σ(x)]e (11) of both electromagnetic and weak interactions based Independently of whether it is expanding around a on a gauge theory with group SU(2)×U(1) but, in or- particular vacuum for the scalar field it is to keep in der to reproduce Fermi’s four-fermions theory of the mind that the whole Lagrangian is still gauge invariant β-decay it was necessary that two of the vector fields under (9). This means that performing a gauge trans- involved would be massive. Also it can be suppose that formation with parameter α(x) = −ϑ(x) can be get rid of in condensed matter physics massive vector fields are the phase in Eq. (11). Substituting then φ(x) = µ + (1/√2) required to describe certain systems, most notably in σ(x) in the Lagrangian it can be find supertransportation. L = −¼FµνFµν+ e2µ2.A Aµ+½∂ σ∂µσ −½λµ2σ2− The way out to this situation is found in the concept µ µ 3 4 4 2 µ 2 µ 2 of spontaneous symmetry breaking. The consistency λµσ −λ σ + e µAµA σ + e AµA σ . (12) of the quantum theory requires gauge invariance, but What are the excitation of the theory around the this invariance can be realized as a Nambu-Goldstone vacuum <φ> = µ? First, it can be finding a massive real relativistic local quantum field system. When this is the scalar field σ(x). The important point however that is case the full gauge symmetry is not explicitly present the vector field A now has a mass given by in the effective action constructed around the particular µ m 2 = 2e2µ2. (13) vacuum chosen by the theory. This makes possible the γ existence of mass terms for gauge fields without taking The remarkable thing about this way of giving a a risk for the consistency of the full theory, which is mass to the photon is that at no point can be given up still invariant under the whole gauge group. gauge invariance. The symmetry is only hidden. There- To illustrate the Higgs mechanism it is possible to fore in quantizing the theory it can still enjoy all the ad- study the simplest example, the Abelian Higgs model: vantages of having a gauge theory but at the same time a U(1) gauge field coupled to a self-interacting charged it have managed to generate a mass for the gauge field. complex scalar field φ with Lagrangian It is surprising, however, that in the Lagrangian (12) it did not found any massless mode. L = −¼F Fµν + ((∂ - ieA )φ)2 −λ-4(|φ|2 − µ2)2, (8) µν μ µ Since the vacuum chosen by the scalar field breaks the where ∂μ - ieAµ is the covariant derivative with e the U(1) generator of U(1) it is to expect one massless parti- electric charge. This theory is invariant under the gauge cle from Goldstone’s theorem. To understand the fate of transformations the missing Goldstone boson it is to revisit the calculation leading to Eq. (12). Where it is to deal with a global φ → eiα(x)φ, A → A + ∂ α(x). (9) µ µ µ U(1) theory, the Goldstone boson would correspond to The minimum of the potential is defined by the equa- excitation of the scalar field along the valley of the poten- tion |φ| = µ. So there is a continuum of different vacua tial and the phase ϑ(x) would be the massless Goldstone labelled by the phase of the scalar field. None of these boson. However it is to keep in mind that in computing vacua, however, is invariant under the gauge symmetry the Lagrangian is managed to get rid of ϑ(x) by shift- The Scalar Particles in Casimir Vacuum State of the Relativistic Quantum Field System in Living Cells 825

ing it into Aµ using a gauge transformation. Actually, T < Tc. However, by nearing consideration it is clear by identifying the gauge parameter with the Goldstone that so one possibility is given by the collective excita- excitation, it is to fixe completely the gauge and the La- tions as by the Casimir vacuum state as solution of the grangian (12) does not have any gauge symmetry left. impulse Schrödinger functional equation. This states A massive vector field has three polarizations: two can be described by the quantum field methods use the transverse ones plus a longitudinal one. In gauging state of the single see scalar as a superposition of the away, the massless Goldstone boson ϑ(x) is transformed ground state and the see excited particles states that is it into the longitudinal polarization of the massive vec- also that his amplitudes are small by way of compari- tor field. In the literature, this is usually expressed say- son with the amplitude of the ground state. The volume ing that the Goldstone mode is “swallowing up” by the V of the total see relativistic quantum scalar field sys- longitudinal component of the gauge field. It is impor- tem will define the distance between the energy levels tant to realize that in spite of the fact that the Lagrang- corresponding to these collective excitations. ian (12) looks different from the one started with not The connection to the radiation of the dark body lost any degrees of freedom. It is started with the two and other properties of the interacting matter with the polarizations of the photon plus the two degrees of relativistic quantum field system is obvious. freedom associated with the real and imaginary com- But in one case to take in account the attraction ponents of the complex scalar field. After symmetry, between two scalars by the presence of the repulsed breaking it is end up with the three polarizations of the boundary conditions must be lead to phenomenon of massive vector field and the degree of freedom of the new significant property: the state of the scalars system real scalar field σ(x). every from them is to be found in a ground state of the The Abelian Higgs model discussed here can be re- relativistic quantum field system must be not obliga- garded as a toy model of the Higgs mechanism respon- tory a state with the smallest energy equally to the en- sible for giving mass to the W± and Z0 gauge bosons in ergy of the “zero fluctuations”. The presence of a few the Standard Model. In condensed matter physics the scalars can lead to lessening of the middle quantity of symmetry breaking described by the relativistic ver- the Casimir energy from the distance between the sca- sion of the Abelian Higgs model as a quantum field lars depended so that this lessening more will compen- system can be used to characterize the onset of a su- sate for the increase of the middle kinetically energy per transportation phase in the BCS theory, where the of the “zero fluctuations” which is connected with the complex relativistic quantum scalar field φ with the coming of those collective states. Casimir vacuum state is associated with the Cooper The supposing that by the absence of the attraction pairs. In this case, the parameter µ2 depends on the between the scalars the ground state total will be a state 2 temperature. Above the critical temperature Tc, µ (T) in which all scalars “are condensate” in a state with   > 0 and there is only a symmetric vacuum <φ> = 0 but impulse k⊥ = q⊥ = 0 and taking in account the small by given boundaries it is to keep in mind the Casimir attraction at the distance, i.e. the Casimir Force, be- force acted at the distance, i.e. the existence of mass- tween the scalars in Casimir vacuum state lead to so less Goldstone particles . When, on the other hand, T < one ground state of the system in which in this case in a 2 Tc then µ (T) < 0 and symmetry breaking takes place. single particle state appear mixture of the excited states The onset of a nonzero mass of the photon (13) below with impulse q3 = k3 ≠ 0. the critical temperature explains the Meissner effect: In 1946 the shift for scalar field ϕ(x) = const + u(x) the magnetic fields cannot penetrate inside supercon- in the theory of microscopically suppertransportation ductors beyond a distance of the order1/mγ. was given by N.N Bogolubov. Conclusion References It is to suppose that is impossible to transport in- Belaus, A. M. and Ts. D. Tsvetkov, 1985. Nauchnie Ocnovi teractions energy by the vacuum state by temperature Tehnology Sublimatsionova Konservanija, Kiev, Nauko- 826 Ts. D. Tsvetkov and G. Petrov

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