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BOUND STATES in QUANTUM FIELD THEORY Conventional Field Theory Is Incomplete

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BOUND STATES in QUANTUM FIELD THEORY Conventional Field Theory Is Incomplete M. GELL —MANN AND F. LOW where E, is the Minkowski force on the particle where T.,= —(1/c')(p(A+ TS)u u, +u.Q +u,Q,7+y., (85) E,= t(k, u4/ic)dV Equation (84) describes motion of a "naked" element, i.e., an element devoid of thermodynamic structure, through a field described by the energy-momentum The mass m though invariant in a Lorentz transforma- tensor T„. If we take T, to be the electromagnetic tion is not necessarily a constant of the motion, even if tensor, the of the particle is constant, since m in- density p T,= (1/4m)(F, Fp,+b„Fp,2/4), (86) cludes the thermodynamic internal energy Lsee Eq. — (34)7. Included in the Minkowski force is a term which where Ii„=8 A, B,A. and A, is the four-vector po- the predicts a force on a particle when it exists in a tempera- tential, then Q, coincides with the Poynting flux, density swhere Z' and H' ture gradient, a force associated with the entropy the energy pU=(E"+EP')/8 are the electric and magnetic intensities in the rest particle possesses. If 8„8,T is constant throughout the frame, and the stress tensor volume of the particle, then this force is —$8„8,T, where 8 is the total entropy of the particle. 41I Q~~ = —(F~p+u~u~F~p/c )(F~p+u~u„F„p/c ) Equation (25) may be written in a form which facili- +8.,F,„2/4, (87) tates comparison with the methods of field theory, which reduces in the rest frame to the maxwell stress namely, tensor in its nine space components and to zero in its ~aP+rr@r = ~~rTorp (84) fourth row and column. PH YSI CAL REVI EW VOLUM E 84, NUM B ER 2 OCTOBER 15, 1951 Bound States in Quan@~~ Field Theory MURRAY GELL-MANN AND FRANcIs Low Institute for Advanced Study, Princeton, Eem Jersey (Received June 13, 1951) The relativistic two-body equation of Bethe and Salpeter is derived from Geld theory. It is shown that the Feynman two-body kernel may be written as a sum of wave functions over the states of the system. These wave functions depend exponentially on the energies of the states to which they correspond and therefore provide a means of calculating energy levels of bound states. L INTRODUCTION Schroedinger equation for the state vector with the EVERAL attempts have been made to calculate the requirement that it contain no particle-pairs and only one field quantum. The formal extension of his method energy levels of bound systems of two particles that is on interact through a quantized field. The standard to include higher approximations dificult account nonco- method' has been to calculate an efFective potential of the necessity of separating divergences in a variant Furthermore, it impossible in his energy function and to insert that function into some way. appears framework to make use of the elegant techniques two-particle Schroedinger or Dirac equation. In a case ' where the major efFects of the interaction are obviously developed by Feynman4 and Dyson. and have an equation' for in the nonrelativistic region (e.g., the hydrogen atom), Bethe Salpeter proposed such a procedure seems reasonable (although even here a two-body "wave function"; their equation is covariant higher order efFects may not be describable by a poten- in form and permits the separation of divergences as in tial). ' However, in the treatment of nuclear problems, the S-matrix theory. Their reasoning, however, is based one may have to deal with specifically relativistic inter- on an analogy to that in Feynman's "Theory of actions and singular forces for which methods successful positrons"4 and the demonstration of equivalence to in the atomic domain may fail entirely. Recently, DancofP has used an approximate method 4 R. P. Feynman, Phys. Rev. 76, 749 (1949);Phys. Rev. 76, 769 (1949). based directly on field theory. He has solved the ~ F. J. Dyson, Phys. Rev. 75, 486 (1949); Phys. Rev. 75, 1736 (1949). ' G. Breit, Phys. Rev. 34, 553 (1929);Yukawa, Sakata, Koba- s H. A. Bethe and E. E. Salpeter, Phys. Rev. 82, 309 (1951). yashi, and Taketani, Proc. Phys. -Math. Soc. Japan 20, 720 (1938). We are indebted to Drs. Bethe and Salpeter for communicating ' Y. Nambu, Prog. Yheor. Phys. 5, 614 (1950). their results to us prior to publication. We understand that this 'S. M. Dancoff, Phys. Rev. 78, 382 (1950). equation has been treated by Schwinger in his lectures at Harvard. BOUND STATES IN QUANTUM FIELD THEORY conventional field theory is incomplete. Our purpose is We shall make extensive use of the true vacuum state to provide such a demonstration. 0'p and the vacuum of free particles, @p. The former is The starting point of the argument of Bethe and the eigenstate of H with the lowest energy pop= Ep. The Salpeter is the consideration of the so-called Feynman latter is the eigenstate of Ho=Ho(0) with the lowest two-body kernel, and the proof that its usual power eigenvalue ep. series expansion can be re-expressed as an integral It will be necessary to express 0 p in terms of C p and equation. We will begin in a similar fashion; however, operators of the interaction representation. There are we will exhibit all quantities of which we make use (in two standard methods of handling that problem. The particular, the wave function) as matrix elements of first is to introduce the interaction representation at a field operators, and derive their properties in the con- time 3= —~ rather than at t=0 as we have done; it is ventional way. then assumed that at —~ the coupling constant vanishes, H and Hp are identical, and thus 0 p=Cp. It II. GENERAL THEORY will become clear that such a procedure does indeed For simplicity we will consider a proton 6eld f and a yield correct results, but it would be inconsistent to neutron field p coupled by a neutral scalar meson field, base a discussion of stationary states on a physical A. {The generalization to other cases is direct. ) All assumption of the variability of charge. The second symbols unless otherwise labeled refer to the Heisenberg method is that of stationary-state perturbation theory; representation. Let us denote by I'„ the total energy- the interaction representation can be defined as above momentum four vector of the interacting fields; P„acts at I,=O, and 4p expressed in the form, — as a displacement operator in the sense that for any c%'p —[1+(Hp—pp) '(1—V)(Hr(0) —Eo+pp)] 'Cp, (9) function I of the field-variables at the space-time point x the equation where V is the projection operator on the state C p and 4IF/Bx„=i[F, P„j (1) c is a normalization constant. We are then free to write Eq. (9) in terms of integrals over time (as a parameter), holds. ' %'e may choose a complete set 0' of state- for example in terms of the U matrix considered as a vectors such that solution of Eq. (7). We shall show in the Appendix that P„4.=p„"4.. (2) Eq. (9) may be replaced by the formal equation„ — Each 4' will then describe a stationary state. c%'o=U '(&Op, o)Co/(Co, U '(%~, 0)Co). (10) I'p The component can be written as the hamiltonian In fact, both the numerator and denominator of Kq. function H of the field variables and their conjugate (10) are indeterminate on account of the presence of a momenta. is it is Although H independent of time, phase factor exp(i~); however, the quotient is well convenient two time-dependent to separate it into defined in terms of a suitable limiting process, such as and where is the sum the parts, H, (t) Hr(t), Ho(t) of we shall exhibit in the Appendix, which does indeed free-field hamiltonians. We shall use IJp to define an involve turning the charge on and off infinitely slowly, interaction reduces the Heisen- representation that to but only as part of a mathematical prescription for berg representation at a finite time, which we shall take solving the stationary state problem. to be t=o For any tim. e dependent operator 0(t) the corresponding operator 0(t) in the interaction repre- III. THE INTEGRAL EQUATION FOR THE KERNEL sentation is given by We define the Feynman two-body kernel as follows: 0(t) = exp(iHp(0)t)0(0) exp( —iHp(0)t). (3) E(x~xo, xpx4) = (%p„P[$(x )$4(xo)f(xp) y(x4)]ep), (11) According to Eq. (1), the time-dependence of the where I' is Dyson's time-ordering operator and e is —1 Heisenberg operators is of the form, if the permutation of the times (1234) induced by P = — is even, +1 if it is odd. Introducing the interaction 0(t) exp(iHt) 0(0) exp( iHt) (4) representation at time t= 0, and making use of Eqs. (5) so that Eq. (3) can be written and (10), we have, for a typical order of the times (t, &t,&t, &t4), 0(t) = U(t 0)0(t)U-'(t, 0), (5) where X(x,x„x~4)= —(C p, U(+ ~, O) U-4(t„O) q(x,) — ~ U(t, 0)=exp(iHp(0)t) exp( iHt), X U(t„o)U-'(t„o) tI(x,) " and thus satisfies the familiar diBerential equation, X P(x4) U(t4, 0)U-'( —~, 0)C p)/ id U(t, 0)/dt= H, (t) U(t, 0), (7) (C'o) U(~, 0)U-'( —~, o)C'o), (12) with the boundary condition where the ratio is necessary to eliminate the constant c, and terms of the form (C'o, U '(& ~, 0)Cp) have can- U(0, 0) =1.
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