Eliminating ultraviolet divergence in quantum field theory through use of the Boltzmann factor
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Eliminating ultraviolet divergence in quantum fieldtheory through use of the Boltzmann factor
Kohzo
Nishida ∗ ) Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan
Abstract
The need for a cutoff in the Lamb shift calculation suggests that high-energyvirtual photons do not interact with real particles. In this paper, we assume thatthe creation of high-energy virtual particles is suppressed by a Boltzmann factor. Asa result, the Coulomb potential is modified, and the zero-point energy density andone-particle-irreducible self-energy of the scalar field are finite. ∗ ) E-mail: [email protected] typeset using PTP
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1. Introduction
In quantum field theory, cutoffs are often introduced when calculating physical quantities.Let us show this in the Lamb shift calculation. The Lamb shift ∆E between the 2 s / and 2 p / levels is known to be ∆E = α m e π Z m e /a dpp = α m e π ln 1 α , (1)where m e is the mass of the electron, α is the fine structure constant, and a = 1 / ( m e α ) is theBohr radius. The energy integral of the virtual photon, R dp/p , is stops counting the photonswhen their wavelength gets bigger than the size of the atom, a . On the short wavelengthside, this integral stop counting the photons when their wavelength gets shorter than theCompton wavelength. That is, the cutoff m e is introduced in the Lamb shift calculation.Renormalization by cutoff is also an operation that does not count virtual particles withenergy larger than the cutoff. Thus, the agreement between the experimental value and thetheoretical value introducing the cutoff suggests that high-energy virtual photons do notvirtually affect real particles.It is known that equation (1) with a cutoff m e can be approximated to an integral withan infinite range by using a suppression factor e − p/m e as follows ∆E ≃ α m e π Z ∞ /a dpp e − p/m e ≃ α m e π (cid:18) ln 1 α − γ + πα (cid:19) , (2)where we used the integral formula Z ∞ x e − p p dp = Γ (0 , x ) = ( − log( x ) − γ ) + x − x · · · , (3)where Γ ( s, x ) is the upper incomplete gamma function, and γ = 0 . · · · is Euler’sconstant. Furthermore, we can approximate the integrals I = Z Λ p n dk = 1 n + 1 Λ n +1 ( n = 0 , , · · · ) (4)in the loop integral as follows when n is small: I ≃ Z ∞ p n e − p/Λ dp = n ! Λ n +1 . (5)Thus, in general, we can approximate an integral with a cutoff Λ as an integral with aninfinite integral range multiplied by the suppression factor e − p/Λ .The suppression factor e − p/Λ has the same form as a Boltzmann factor e − βE . If thesuppressor is a Boltzmann factor, it means that the creation of virtual particles follows acanonical distribution. This study aims to investigate how the Boltzmann factor affectsquantum field theory. 2
2. Modified Coulomb potential
The momentum representation of the Coulomb potential is V ( r ) = − Ze (2 π ) Z ∞−∞ d pe − i p · r | p | , (6)where Z is the atomic number. This potential includes the contribution of high-energyvirtual photons. If the creation of the high-energy virtual photons is suppressed by theBoltzmann factor, equation (6) should be modified as follows: V ( r ) = − Ze (2 π ) Z ∞−∞ d pe − i p · r | p | e −| p | /m c , (7)where m c is a constant with mass as a dimension. This polar coordinate expression is V ( r ) = − Ze (2 π ) Z ∞ dp Z π dθ Z π dφp sin θe − ipr cos θ p e − p/m c = − Ze (2 π ) πr Z ∞ dp sin prp e − p/m c , (8)where p = | p | . Using the integral formula Z ∞ dx sin rxx e − x/m = arctan( mr ) , (9)we finally obtain the modified Coulomb potential: V ( r ) = − Ze π r arctan( m c r ) . (10)Using arctan( x ) = x − x / · · · and lim x →∞ arctan( x ) = π/
2, (10) can be approximatedin each region as follows: V ( r ) = − Ze m c π + Ze m c π r ( m c r ≪ , − Ze πr ( m c r ≫ . (11)This potential has an interesting property that it is an ordinary Coulomb potential at longrange and becomes finite at r = 0. §
3. Modification of quantum field theory
In this section, we consider a quantum field theory of scalar fields with the Boltzmannfactor. In the Fourier transforms of position and momentum space, we request that a rela-tivistic Boltzmann factor be multiplied as follows: f ( x ) = Z ∞−∞ d p p (2 π ) F ( p ) e i p · x × e −| ¯ u µ p µ ( p ) | / ( am c ) (12)3or position space and F ( p ) = Z ∞−∞ d x p (2 π ) f ( x ) e − i p · x × e | ¯ u µ p µ ( p ) | / ( am c ) (13)for momentum space, where a = 1 or 2, m c is a constant with mass as a dimension, and p µ ( p ) ≡ ( ω ( p ) , p ). ¯ u µ is an average of four-velocities of particles created by the scalar field φ . e | ¯ u µ p µ ( p ) | / ( am c ) in (13) shifts the origin of distribution function like chemical potential.The spatial distribution of matter in the universe is homogeneous and isotropic, whichmeans that the average created particle four-velocities ¯ u µ can be written in any coordinatesystem as ¯ u µ = (1 , , , , (14)because the spatial component is canceled with plus and minus appearing equally. So in anycoordinate system, we can always rewrite the the relativistic Boltzmann factor to e −| ¯ u µ p µ ( p ) | / ( am c ) = e − ω ( p ) / ( am c ) . (15)From (12) with a = 2, a scalar field is expanded as follows: φ ( t, x ) = Z d p p (2 π ) ω ( p ) { a ( p ) e − ipx + a † ( p ) e ipx } e −| u µ p µ ( p ) | / (2 m c ) . (16)The Lagrangian density of the simple free scalar field is given by L = 12 ∂ µ φ ( x ) ∂ µ φ ( x ) − m ( φ ( x )) . (17)The Hamiltonian is given by H = Z d x { ( ˙ φ ( x )) − L} . (18)Substituting (16) into the Hamiltonian (18), we have H = Z d pω ( p ) (cid:26) n ( p ) + 12 V ∞ (2 π ) (cid:27) e −| ¯ u µ p µ ( p ) | /m c , (19)where V ∞ ≡ R d x and n ( p ) = a † ( p ) a ( p ) is a number operator with momentum p . Therefore,the zero-point energy density is calculated as < | H | >V ∞ = Z ∞−∞ d p (2 π ) ω ( p ) e −| ¯ u µ p µ ( p ) | /m c = Z ∞−∞ d p (2 π ) ω ( p ) e − ω ( p ) /m c m c (2 π ) Z ∞ A y p y − A e − y dy ( y = p /m c + A , A = m/m c )= m c π ) ((cid:18) mm c (cid:19) K (cid:18) mm c (cid:19) − (cid:18) mm c (cid:19) K (cid:18) mm c (cid:19)) ≤ m c (2 π ) Z ∞ A y e − y dy = m c e − m/m c (2 π ) ((cid:18) mm c (cid:19) + 3 (cid:18) mm c (cid:19) + 6 mm c + 6 ) , (20)where we used (14). The equal sign can be used only when m = 0. K ( x ) and K ( x ) are themodified Bessel function of the second kind. We obtained a finite zero-point energy density.Using (16), the Feynman propagator function is calculated, ∆ F ( x, y )= − i < | T ( φ ( x ) φ ( y )) | > = − i Z d p p (2 π ) ω ( p ) Z d q p (2 π ) ω ( q ) e −| ¯ u µ p µ ( p ) | / (2 m c ) e −| ¯ u µ q µ ( q ) | / (2 m c ) × [ < | θ ( x − y ) a ( p ) e − ipx a † ( q ) e iqy + θ ( y − x ) a ( q ) e − iqy a † ( p ) e ipx | > ]= − i Z d p (2 π ) ω ( p ) e −| ¯ u µ p µ ( p ) | /m c ×{ θ ( x − y ) e − ip ( x − y ) + θ ( y − x ) e ip ( x − y ) }| p = ω ( p ) , (21)which can be rewritten as ∆ F ( x, y ) = Z d p (2 π ) e − ip ( x − y ) p − m + iǫ e −| ¯ u µ p µ ( p ) | /m c (22)using ω ( − p ) = ω ( p ).Consider the Feynman propagator function in momentum-space: i∆ F ( p ) ≡ Z d xe ipx i∆ F ( x, e | ¯ u µ p µ ( p ) | /m c = Z d q (2 π ) iq − m + iǫ e −| ¯ u µ q µ ( q ) | /m c (2 π ) δ ( q − p ) e | ¯ u µ p µ ( p ) | /m c = ip − m + iε . (23)Since (23) is the Fourier transform of ∆ F ( x, a = 1.Equation (23) picking out the specific energy means that the origin of canonical distribu-tion is shifted by u µ p µ ( p ). Similarly, we define the one-particle state with specific momentum5 as | p > ≡ e | u µ p µ ( p ) | / (2 m c ) a † ( p ) | > . (24)Then, we find that the amplitude associated with an external leg with p is < | φ ( x ) | p > = e − ipx p (2 π ) ω ( p ) . (25)Eventually, from (23) and (25), we find that the Boltzmann factor e −| ¯ u µ p µ ( p ) | /m c has no effecton the Feynman rules at the tree level (zero-loop).Next, consider the case where there is an interaction term L int = − λ φ . For a first-ordertwo-point function, we have G (2)1 ( x , x ) ≡ − iλ Z d y [ i∆ F ( x , y )][ i∆ F ( y, y )][ i∆ F ( x , y )]= Z d y Z d p (2 π ) ie − ip ( x − y ) p − m + iǫ e −| ¯ u µ p µ ( p ) | /m c × − iλ Z d l (2 π ) ie −| ¯ u µ l µ ( l ) | /m c l − m + iǫ × Z d p (2 π ) ie − ip ( x − y ) p − m + iǫ e −| ¯ u µ p µ ( p ) | /m c = Z d p (2 π ) ie − ip x p − m + iǫ e −| ¯ u µ p µ ( p ) | /m c × − iλ Z d l (2 π ) ie −| ¯ u µ l µ ( l ) | /m c l − m + iǫ × Z d p (2 π ) ie − ip x p − m + iǫ e −| ¯ u µ p µ ( p ) | /m c (2 π ) δ ( p + p )= Z d p (2 π ) ie − ip ( x − x ) p − m + iǫ e −| ¯ u µ p µ ( p ) | /m c × − iλ Z d l (2 π ) ie −| ¯ u µ l µ ( l ) | /m c l − m + iǫ × ip − m + iǫ e −| ¯ u µ p µ ( p ) | /m c . (26)Using the modified Fourier transform (13) with a = 1, we obtain the two-point Green’sfunction in momentum-space i∆ ′ F ( p ) = Z d xe ipx { i∆ F ( x,
0) + G (2)1 ( x,
0) + · · · } e | ¯ u µ p µ ( p ) | /m c = ip − m + iǫ + ip − m + iǫ {− iΣ ( p ) } ie −| ¯ u µ p µ ( p ) | /m c p − m + iǫ + · · · = ip − m − Σ ( p ) e −| ¯ u µ p µ ( p ) | /m c + iǫ , (27)where − iΣ ( p ) is the one-particle-irreducible self-energy function − iΣ ( p ) = − iλ Z d l (2 π ) il − m + iǫ e −| ¯ u µ l µ ( l ) | /m c . (28)6et us calculate Σ ( p ) in (28). We have Σ ( p ) = λ Z d l (2 π ) il − m + iǫ e −| ¯ u µ l µ ( l ) | /m c = iλ Z d l (2 π ) e −| ¯ u µ l µ ( l ) | /m c Z C dl l − ω + iǫ )( l + ω − iǫ )= iλ Z d l (2 π ) e −| ¯ u µ l µ ( l ) | /m c πil − ω + iǫ (cid:12)(cid:12)(cid:12)(cid:12) l = − ω + iǫ = λ Z ∞−∞ d l (2 π ) ω ( l ) e −| ¯ u µ l µ ( l ) | /m c . (29)Again, using ¯ u µ = (1 , , , Σ ( p ) = λ Z ∞−∞ d l (2 π ) ω ( l ) e − ω ( l ) /m c = λm c π ) π Z ∞ A p y − A e − x dy ( A = m/m c )= λm c π mm c K (cid:18) mm c (cid:19) ≤ λm c π Z ∞ A xe − y dy = λm c e − m/m c π (cid:18) mm c + 1 (cid:19) , (30)where the equal sign can be used only when m = 0. K ( x ) is is the modified Bessel functionof the second kind. We obtained a finite self-energy.Finally, let us give one of the four-point vertex correction to see an example of theBoltzmann factor: i ˜ Γ (4) (( p + p ) )= ( − iλ ) Z d l (2 π ) ie −| ¯ u µ l µ ( l ) | /m c l − m + iǫ × ie −| ¯ u µ ( p µ ( p )+ p µ ( p ) − l µ ( l )) | /m c { ( p + p ) − l } − m + iǫ . (31) §
4. Conclusion
In the first section, we have shown that the introduction of cutoff into theory is equivalentto the introduction of the Boltzmann factors. Next, we derived a non-relativistic Coulombpotential in which high-energy virtual photons are suppressed by the Boltzmann factor.Furthermore, we tried to introduce the relativistic Boltzmann factor into quantum fieldtheory, and demonstrated that the factor results in finite values for the zero-point energydensity and one-particle-irreducible self-energy of the scalar field.7 eferences