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Register a new userLagrangian quantum field theory in momentum picture. IV. Commutation relations for free fields
نظرية الحقل الكمي لاكرانجي في الصورة الحذفية. IV. العلاقات التراكمية للحقول الحرة
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تُناقش من وجهة نظر رياضية علاقات التبادل الجبرية المحتملة في نظرية لاقونية الكم الحر للحقول المقياسية والإشارية والمتجهة. كمصادر لهذه العلاقات تستخدم المعادلات/العلاقات الهايزنبرج للمتغيرات الديناميكية وشرط محدد للفردية لمشغلات المتغيرات الديناميكية (مقابل بعض الطبقات من اللاقونيات). وتشير العلاقات الجانبية التبادلية أو بعض تطويراتها إلى أنها الأكثر عامة والتي تؤدي إلى صحة كل معادلات هايزنبرج. يصبح الامتثال لمعادلات هايزنبرج وشرط الفردية أمراً مستحيلاً. ويتم حل هذه المشكلة من خلال إعادة تعريف المتغيرات الديناميكية، مماثلة لعملية الترتيب الطبيعي والتي تشملها كحالة خاصة. وهذا يؤدي إلى تغييرات مقبولة في العلاقات التبادلية. وتضيف دخول مفهوم الفراغ إلى الطبقة المحتملة للعلاقات التبادلية؛ وبالخصوص، يقتضي ذلك تخفيض إعادة تعريف المتغيرات الديناميكية إلى الترتيب الطبيعي. ويتم تطبيق آخر قيد على هذه الطبقة من خلال الطلب عن وجود إجراء تحليلي فعال لحساب القيم المتوسط للفراغ. وتشير العلاقات التبادلية المثلثية القياسية إلى الوحيدة المعروفة التي تلبي كل من الشروط المذكورة ولا تتناقض مع البيانات الحالية.
Possible (algebraic) commutation relations in the Lagrangian quantum theory of free (scalar, spinor and vector) fields are considered from mathematical view-point. As sources of these relations are employed the Heisenberg equations/relations for the dynamical variables and a specific condition for uniqueness of the operators of the dynamical variables (with respect to some class of Lagrangians). The paracommutation relations or some their generalizations are pointed as the most general ones that entail the validity of all Heisenberg equations. The simultaneous fulfillment of the Heisenberg equations and the uniqueness requirement turn to be impossible. This problem is solved via a redefinition of the dynamical variables, similar to the normal ordering procedure and containing it as a special case. That implies corresponding changes in the admissible commutation relations. The introduction of the concept of the vacuum makes narrow the class of the possible commutation relations; in particular, the mentioned redefinition of the dynamical variables is reduced to normal ordering. As a last restriction on that class is imposed the requirement for existing of an effective procedure for calculating vacuum mean values. The standard bilinear commutation relations are pointed as the only known ones that satisfy all of the mentioned conditions and do not contradict to the existing data.
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