Advertisement

Il Nuovo Cimento A (1965-1970)

, Volume 70, Issue 1, pp 1–11 | Cite as

On current-field identities

  • H. Genz
  • J. Katz
Article
  • 17 Downloads

Summary

The spectral representation for the vacuum expectation value of the axial-vector-current commutator is used to discuss the current-field identities given byg A a μ α (x)=fπμϕα(x)+A μ α (x). Herea μ α (x) denotes the axial-vector-meson field,gA is a properly chosen normalization constant, and the pion field ϕα(x) is defined by the relation\(\partial ^\mu A_\mu ^\alpha (x) \equiv \widehat\varphi ^\alpha (x) = f_\pi m_\pi ^2 \varphi ^\alpha (x)\), whereA μ α (x) denotes the axial vector current. We find that
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
and that these expressions vanish if and only if the pion field ϕα(x) is a free field. We also note that
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (\mathop x\limits_r ),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
and that these expressions vanish if and only if the axial vector currentA μ α (x) is conserved. The consequences for canonical realizations of current-field identities and PCAC are given. We also find that the vacuum expectation value of the σ-term is nonvanishing unless the current is conserved and that a nonconserved charge cannot annihilate the vacuum (as stated by Coleman's theorem).

Keywords

Spectral Function Vacuum Expectation Derivative Coupling Axial Vector Current Pion Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

О ток-полевых тождествах

Резюме

Спектральное представление для вакуумной ожидаемой величины коммутатора аксиально-векторного тока используется для обсуждения ток-полевых тождеств, заданных соотношениемg A a μ α (x)=fπμϕα(x)+A μ α (x). Здесьa μ α (x) обозначает поле аксиально-векторного мезона,gA есть надлежашим образом выбранная перенормировочная постоянная; пионное поле ϕα(x) определяется соотношением\(\partial ^\mu A_\mu ^\alpha (x) \equiv \widehat\varphi ^\alpha (x) = f_\pi m_\pi ^2 \varphi ^\alpha (x)\), гдеA μ α (x) представляет аксиально-векторный ток. Мы получаем, что
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
и что эти выражения обрашаются в нуль, если и только если пионное поле ϕα(x) представляет свободное поле. Мы также отмечаем, что
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (\mathop x\limits_r ),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
и что эти выражения обращаются в нуль, если и только если аксиально-векторный токA μ α (x) сохраняется. Приводятся следствия для канонических реализаций токполевых тождеств и РСАС. Мы также находим, что вакуумная ожидаемая величина σ-члена не обращается в нуль, если ток не сохраняется, и что несохраняющийся заряд не может уничтожить вакуум (как утверждается теоремой Колемана).

Riassunto

Si usa la rappresentazione spettrale del valore di aspettazione del vuoto per discuter e le identità corrente-campo date dag A a μ α (x)=fπμϕα(x)+A μ α (x). Quia μ α (x) denota il campo del mesone vettoriale assiale,gA è una costante di rinormalizzazione opportunamente scelta, ed il campo pionico ϕα(x) è definito dalla relazione\(\partial ^\mu A_\mu ^\alpha (x) \equiv \widehat\varphi ^\alpha (x) = f_\pi m_\pi ^2 \varphi ^\alpha (x)\), doveA μ α (x) denota la corrente vettoriale assiale. Si trova che
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
e che queste espressioni si annullano se e solo se il campo pionico ϕα(x) è un campo libero. Si nota anche che
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (\mathop x\limits_r ),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
e che queste espressioni si annullano se e solo se la corrente vettoriale assialeA μ α (x) è conservata. Si espongono le conseguenze per le realizzazioni canoniche delle identità corrente-campo e per PCAC. Si trova anche che il valore di aspettazione del vuoto del termine σ non tende a zero a meno che la corrente sia conservata e che una carica non conservata non possa annichilare il vuoto (come è affermato dal teorema di Coleman).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. (1).
    R. Arnowitt, P. Nath andM. H. Friedman:Phys. Rev. Lett.,22, 1158 (1969);P. Nath, R. Arnowitt andM. H. Friedman:Veneziano amplitude, current algebra and vertex functions, preprint, Northeastern University, Boston.ADSCrossRefGoogle Scholar
  2. (2).
    T. D. Lee andB. Zumino:Phys. Rev.,163, 1667 (1967);J. Wess andB. Zumino:Phys. Rev.,163, 1727 (1967), and references therein.ADSCrossRefGoogle Scholar
  3. (3).
    R. Arnowitt, M. H. Friedman andP. Nath:Nucl. Phys.,10 B, 578 (1969), and references therein.ADSCrossRefGoogle Scholar
  4. (4).
    R. Acharya, P. Narayanaswamy andT. S. Santhanam:A rigorous lower bound on the KSFR relation from field theory, Internal Report IC/69/63, ICTP, Trieste.Google Scholar
  5. (5).
    Statement II has been partly derived inH. Genz:Zeits. Phys.,229, 206 (1369).CrossRefGoogle Scholar
  6. (6).
    It also follows from this (ref. (5)H. Genz:Zeits. Phys.,229, 206 (1369)) that the σ-term has a nonvanishing vacuum expectation value and thus a nonconserved charge cannot annihilate the vacuum as stated by Coleman's theorem (ref. (7) ).ADSMathSciNetCrossRefGoogle Scholar
  7. (7).
    S. Coleman:Phys. Lett.,19, 144 (1965);Journ. Math. Phys.,7, 787 (1966).ADSMathSciNetCrossRefGoogle Scholar
  8. (8).
    T. D. Lee, S. Weinberg andB. Zumino:Phys. Rev. Lett.,18, 1029 (1967).ADSCrossRefGoogle Scholar
  9. (8a).
    K. G. Wilson:Phys. Rev.,179, 1499 (1969).ADSMathSciNetCrossRefGoogle Scholar
  10. (8b).
    P. De Mottoni andH. Genz:Nuovo Cimento,67 B, 1 (1970). In the meantime, it was derived thatE 1 has a nonvanishing VEV:P. De Mottoni andH. Genz:On logarithmic behavior in short-distance field products, preprint.ADSCrossRefGoogle Scholar
  11. (9).
    For a proof seeR. Jost:Properties of Wightman Functions, inLectures on Field Theory and the Many-Body Problem, edited byE. R. Caianello (New York, 1961).Google Scholar
  12. (11).
    Clearly, ifj π(0)|Ω〉 vanishes then 〈z|j π(0)|Ω〉 vanishes and thus due to analyticity 〈z 1|j π(0)|z 2〉 also vanishes. However, even though it would still follow that 〈z|j π(0)|Ω〉 is small if ‖j π(0)Ω‖ is small (since all intermediate states given a positive contribution to ‖j π(0)Ω‖) it would no longer necessarily follow that 〈z 1|j π(0)|Z 2〉 is small. The only possible conclusion seems to be that for a particlez that can decay strongly an approximate one-particle saturation implies a relatively long lifetime, since the decay is described by the matrix clement |〈z|j z(0)|Ω〉|, which is small if ‖j z(0)Ω‖ is small.Google Scholar
  13. (12).
    S. Weinberg:Phys. Rev. Lett.,18, 507 (1967). (Although in this reference the Weinberg sum rules were both formulated and derived for conserved currents only, one could use the Jacobi identity as in ref. (13)S. L. Glashow, H. J. Schnitzer andS. Weinberg:Phys. Rev. Lett.,19, 139 (1967) to obtain these sum rules also for nonconserved currents if one makes some assumptions about equal-time commutators and double commutators.)ADSCrossRefGoogle Scholar
  14. (13).
    S. L. Glashow, H. J. Schnitzer andS. Weinberg:Phys. Rev. Lett.,19, 139 (1967);R. Jackiw:Phys. Lett.,27 B, 96 (1968);H. Genz:Spectral Function Sum Rules from Identities of the Jacobi Type, inBoulder Lectures in Theoretical Physics 1969, to be published.ADSCrossRefGoogle Scholar
  15. (15).
    (K. Pohlmeyer:Commun. Math. Phys.,12, 204 (1969).ADSMathSciNetCrossRefGoogle Scholar
  16. (16).
    We assume that all the equal-time limits considered exist so that we may interchange limiting producedures (differentiation, equal-time limits) with the mass intergration in the spectral representations. For a general discussion of equal-time limits and their possible nonexistence in perturbation theory we refer the reader to ref. (17).ADSCrossRefGoogle Scholar
  17. (17).
    J. Katz andJ. Langerholc:Phys. Rev.,184, 1577 (1969).ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1970

Authors and Affiliations

  • H. Genz
    • 1
  • J. Katz
    • 2
  1. 1.Lawrence Radiation LaboratoryUniversity of CaliforniaBerkeley
  2. 2.Purdue UniversityLafayette

Personalised recommendations