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Dirac Equation and Planck-Scale Quantities

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Dirac Equation and Planck-Scale Quantities Dirac Equation and Planck-Scale Quantities Rainer Collier Institute of Theoretical Physics, Friedrich-Schiller-Universität Jena Max-Wien-Platz 1, 07743 Jena, Germany E-Mail: [email protected] Abstract. This paper investigate whether quantities with Planck dimensions occur already in the common quantum theory with local Lorentz symmetry . As the quantities LPl and M Pl involve the Planck constant , the 2 velocity of light c and the gravitational constant G ( MPl = cG, LPl= GM Pl c ), the relativistic Dirac equation ( , c ) in the Newtonian gravitational potential ( G ) is considered as a test theory. The evaluation of the break-off condition for the power series of the radial energy eigenfunctions of this purely gravitational atom leads to exact terms for the energy eigenvalues En for various special cases of the quantum numbers N , k and nN= + | k |. It turns out that a useful model of an atom, based solely on Newtonian gravitational forces, can result if, inter alia, the test mass m0 in the gravitational field of the mass M satisfies the condition mM0 ≤ Pl . 1 Introduction In the last few decades, the terms of Planck length and Planck energy have frequently turned up in the literature on quantum gravity. It is believed that, in spatial ranges of the linear 32 extension L LPl = G c = GMPl c , the space-time manifold is of a grainy or at least no longer continuous structure. Various approaches to a theory of quantum gravity, such as the string theory, loop quantum gravity, non-commutative geometries, approaches to deformed Lorentz and Poincaré algebras, doubly special relativity (DSR) and diverse generalized uncertainty principles (GUP), therefore, attempt to incorporate into the structure of the theory an elementary length that is independent of the observer and proportional to the Planck length. For an overview of this subject area, see [1], [2] and [3]. The present treatise will look into the question in what way physical quantities with Planck dimensions can occur already in the common quantum theory with local Lorentz symmetry. Since in the Planck quantities LPl and M Pl there occur Planck’s constant , the velocity of light c and Newton’s gravitational constant G , the relativistic Dirac equation (with and c ) in the Newtonian gravitational potential (with G ) can be considered as a suitable example. 1 2 The Dirac equation with an electromagnetic field For a better overview of the symbols and terms used, let us first look at the Dirac equation with an electromagnetic field in the flat Minkowski space (with the metric signature and the definition of the four-quantities following the textbook [15]), q γψk p−= mc 0 , p =− P A , P =∂ i . (2.1) k 0 kkc k k k k k k Here, γ denotes the Dirac matrices, p= ( Ecp ,) the kinetic four-momentum, P the k k canonical four-momentum, Aa= (,)ϕ the four-potential, x= ( ct ,) r the four-coordinates k k and ∂k =∂∂x =∂∂( ( ct ), ∇ ) the four-gradient. The γ matrices satisfy the common commutation rules γγk l+= γγ l k2 η kl , (2.2) with η kl =diag (1, −−− 1, 1, 1) denoting the Minkowski tensor. Let the Latin indices kl, ,.... run from 0 to 3, but the first letters ab, ,.... from 1 to 3 only. 2.1 The Dirac equation in the Coulomb potential If we have a positive electric charge Q= Zee 0 situated fixed at the coordinate origin, and a negative electric charge qe= − 0 at a distance of rr= ||, the electric interaction potential Uel (in the Gaussian cgs system) has the form 2 qQ Ze e0 Q Uqel ===ϕ −= , Ze , (2.3) rr e0 where ee0 = || is the absolute value of the electric elementary charge. With the corresponding four-potential k Q Aa=(ϕϕ , ) = ( ,0,0,0) , ϕ= , (2.4) r the special-relativistic Dirac equation then reads q [γψk (P−− A ) mc ] = 0 (2.5) kkc 0 and, with (3+1) splitting, q [()γ0 P− A +− γψa P mc] = 0 . (2.6) 00c a 0 2 With the well-known definitions γβ0 = and βγaa= a , and with definitions (2.1) and (2.2) being considered, multiplication by cβ (in 2.6) yields a 2 [−ca Pa + cP00 −− q ϕβ mc ] ψ = 0 , (2.7) 2 [ca⋅+− p E qϕβ − mc0 ] ψ = 0 . (2.8) With Eiψψ→ and piψψ→∇() , the Dirac equation shows the Hamiltonian structure 2 iψ =[ ca ⋅+ p qϕβ + mc0 ] ψ ≡ H ψ . (2.9) With the ansatz ψχ= (xa ) exp( Et i ) (2.10) 2 there follows the time-independent equation of the energy eigenvalues ()E00= mc , Eχ=[ c a ⋅+ pqϕβ + E0 ]χ . (2.11) With the aid of the Dirac matrices (σ Pauli matrices ) 01 00 σσ 00 0 γ = , γ= , a= γγ= , β= γ (2.12) 0−−1 σσ 00 and, by decomposing the bispinor χ= (, χχG KT ), we obtain from (2.11) the set of equations for determining the energy eigenvalues E , KG cσ⋅ pχ +( Uel −+ EE0 )χ = 0 , (2.13) GK cσ⋅ pχ +( Uel −− EE0 )χ = 0 . (2.14) GK With Uel from (2.3), pi=−∇ and the usual power series for the spinors χχ, , there results the known break-off condition of the radial energy eigenfunctions of the electric atom 22− +− 2αα 2 = E0 EN( k()() Zee) Z ee E , (2.15) with which the boundary conditions for χ G and χ K are met. By solving the equation for E , we get the discrete energy eigenvalues En (Sommerfeld’s fine structure formula), 1 22 EE→ = , a=+− NkZ(a ) , (2.16) n Nk,,2 Nk e e ()Zeea 1+ 2 ()aNk, 3 2 e0 Nk=0,1,2,3,.... , =±±±1, 2, 3,.... , ae = . (2.17) c Here, N is the radial quantum number, kj=±+( 12) with j the total angular momentum quantum number, nN= + || k the principal quantum number, and ae the fine structure constant. 2.2 Evaluation of the electric break-off condition The lowest energy level of a quantum system is known to be its ground state. Here, it is the one for N = 0 and k =1, i.e. for n =1, Ee2 E=00 =−=−E1( Za )22 EZ 1 . (2.18) 12 00ee e c Zeea 1+ a0,1 Hence, the binding and ionization energies relate as = − =22 −a −=− Ebind E10 E mc 0( 1( Ze e ) 1) Eioniz . (2.19) Note that, with an atomic number Zee=1a ≈ 137 , the ground-state energy E1 vanishes, and the ionization energy just equals the energy for creating the electron mass m0 . Therefore, according to this simplified atomic model (assuming point masses, neglecting the movement of the atomic nucleus, no radiation corrections,…), atoms with atomic numbers Ze >137 should not exist. In preparation for the gravitational case, let us directly evaluate the break-off condition (2.15) for a number of special cases without using the equation form solved in terms of energy E (2.16). By substitution of x= EE0 , eq. (2.15) takes the form −22 −aa 2 += ⋅ 1xkZ( ()ee N) () Zee x . (2.20) Now we exactly solve (2.20) for the following cases: N →∞ In this case of extreme excitation, we get 2 x→==1 , E E00 mc . (2.21) N = 0 These are radial „ground states“ with any angular momenta, 4 22 2 1−⋅xkZ − ()()eeaa = Z ee ⋅ x , (2.22) kZ22− ()a ()Z a 2 x= ee , EE=1 − ee . (2.23) |k| 0 k 2 Nk= || Equal radial and angular momentum excitation, −⋅22 −αα 2 + = ⋅ 1x( kZ ()||()ee k) Zee x , (2.24) 2 1 22 EZ0 ()eea x=| k | + kZ − (eea ) , E=1 +− 1 . (2.25) 2|k | 2 k 2 Zkeea || Here, we expand eq. (2.20) according to sufficiently small ae , continuing the series up to the 4 order that is proportional to ()ae : 2 1()Zeea 1 4Nk+ || 46 x=−1 − ()+()Zeeαα OZ( ee ) , N+=|| k n , (2.26) 2n24 8( Nk+ ||)|| k 13()ZZαα24n () = −ee +−ee + a 6 EE0 1 24 OZ((ee )) . (2.27) 2n 8 2| kn | (Zka )||< ee We expand eq. (2.20) according to ()Zae values which are sufficiently close to the angular momentum values ||k , EN k2 2 x== +2||k ||(kZ−+aa ) O ||( kZ − ) . (2.28) 22 3 ee (( ee) ) E 22 0 Nk+ ( Nk+ ) (Zkeea )||= From (2.28) we can read the solution for this limit case, EN x = = . (2.29) 22 E0 Nk+ Note that all these special formulae can also be obtained by substitution in the exact overall solution for E (formula 2.16). 5 3 The Dirac equation in the Newtonian gravitational potential (Version I) Consider a purely gravitationally bound quantum system consisting of a central mass M and a test mass m0 (with spin 2 ) at a distance of r . Here again, let the mass M that creates the gravitational field be firmly fixed at the coordinate origin, so that Newton’s interaction potential has the form mM Um=φ = − G0 . (3.1) gr 0 r The simplest way to obtain a Dirac equation in the Newtonian gravitational field is via the four-momentum relation generalized for curved spaces having the metric gµν , µν 2 µ µ gµν p p= ( mc00 ) , p= mu . (3.2) Here, uµµ= dx dτ denotes the four-velocity and pµ the four-momentum of a particle having the rest mass m0 . Let the Greek indices µν, ,.... run from 0 to 3, whereas the first letters a,β ,.... run from 1 to 3 only. To find out which component of the momentum pµ can be identified with the energy constant E , let us examine the general-relativistic equation of motion (the geodesic equation) of a test particle of the mass m0 in the gravitational field gµν of the mass M , Dp dp µµ=p uλ = −Γ κλpu =0 , (3.3) Ddττµ; λ µλ κ dp µ =Γ=Γmκ uu λ m uuκλ , (3.4) dτ 0µλ κ 0,κ µλ dpµ m = 0 g uuκλ.
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