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Relativistic quantum mechanics of identical bosons

Релятивистская квантовая механика тождественных бозонов

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Il Nuovo Cimento A (1965-1970)

Summary

We extend the probabilistic interpretation of the relativistic scattering of a charged scalar particle by an external electromagnetic field to a system of several such particles. We find that the use of causal Green functions again demands the specification of boundary conditions for particles at the initial time, and for antiparticles at the final time. The resulting theory is essentially one for a fixed number of particles, which can be found in two states, customarily described as «going forward in time» (particles) and «going backward in time» (antiparticles). This feature allows, and strongly recommends, the use of Dirac's many-time formalism, with one time parameter for each particle. The probabilistic interpretation includes the possibilities of pair creation and annihilation.

Riassunto

Si estende l'interpretazione probabilistica dello scattering relativistico di una particella scalare carica con un campo elettromagnetico esterno a un sistema di molte particelle dello stesso tipo. Si trova che l'uso di funzioni di Green causali richiede ancora la specificazione delle condizioni al contorno per particelle al tempo iniziale, e per antiparticelle al tempo finale. La teoria risultante è essenzialmente una teoria per un numero fisso di particelle, che possono trovarsi in due stati, comunemente descritte come «dirette in avanti nel tempo» (particelle) e «dirette indietro nel tempo» (antiparticelle). Questa caratteristica permette, e ciò ne raccomanda vivamente l'uso, l'impiego del formalismo a più tempi di Dirac, con un parametro temporale per ogni particella. L'interpretazione probabilistica include la possibilità della creazione e distruzione di coppie.

Резюме

Мы распространяем вероятностную интерпретацию релятивистского рассеяния заряженной скалярной частицы внешним электромагнитным полем на систему нескольких таких частиц. Мы находим, что использование причинных гриновских функций снова требует уточнения граничных условий для частиц в начальный момент времени и для античастиц в конечный момент времени. Полученная теория, по существу, представляет теорию для фиксированного числа частиц, которые могут быть обнаружены в двух состояниях, обычно описываемых, как «движение вперед по времени» (частицы) и «движение назад по времени» (античастицы). Эта особенность позволяет, и решительно рекомендует, использовать дираковский много-временной формализм, с одним временным параметром для каждой частицы. Вероятностная интерпретация включает возможности рождения пар и аннитиляции пар.

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  14. The boundaries we consider are spacelike planes in Minkowski space, and we actually have to specify initialand final values.

  15. An equation number such as (1–10) means eq. (10) in ref. (1)E. Marx:Nuovo Cimento,60 A, 669 (1969).

  16. The prime in ∂ μ′ indicates that it operates onx′.

  17. The operatorsP (±)0 are projection operators in the space of solutions of the homogeneous Klein-Gordon equation, in which case they separate the usual positive and negative frequency parts. They satisfy the equations (7′)P (+)0 +P (−)0 =1,\(\begin{gathered} P_0^{( + )} + P_0^{( - )} = 1, \hfill \\ P_0^{( + )} P_0^{( - )} = P_0^{( - )} P_0^{( + )} = (2\tilde E)^{ - 2} (\partial ^2 + m^2 ), \hfill \\ (P_0^{( \pm )} )^2 = P_0^{( \pm )} - (2\tilde E)^{ - 2} (\partial ^2 + m^2 ) \hfill \\ \end{gathered} \),

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  19. We have chosen the probability amplitudes in momentum space so thata (+)(k)=a(k) anda (−)(k)=b *(−k) in the usual notation; they are the component of the Fourier transform ofg(x), which we designate byf(k). We also use an extended summation convention, in which repeated variables (continuous indices), such ask′, are to be integrated over the whole corresponding space.

  20. We use indices 1, 2 and 3 on operators, sometimes in parentheses, to indicate the set of variables on which they operate. For instance,P (±)01 is the operatorP (±)0 of eq. (6) acting on the variablesx 1μ.

  21. These functions have to be properly symmetrized, that is, ϕ(+++ and ϕ(--- are symmetric inx 1,x 2 andx 3, while ϕ(++- is symmetric inx 1 andx 2, and ϕ(+++, inx 2 andx 3.

  22. These indices range over + and −, or over +1 and −1, where appropriate.

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  1. Drexel University, Philadelphia, Pa.

    E. Marx

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  1. E. Marx
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Marx, E. Relativistic quantum mechanics of identical bosons. Nuovo Cimento A (1965-1970) 67, 129–152 (1970). https://doi.org/10.1007/BF02725172

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