Введение
Уравнение поля для спиновых 3/2 фермионов
In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation of spin 3/2 fermions in a four dimensional flat spacetime. It is similar to the Dirac equation for spin 1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941. In modern notation it can be written as:
where is the Levi Civita symbol,
are Dirac matrices (with ) and ,
is the mass,
,
and is a vector valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the representation of the Lorentz group, or rather, its part. This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:
where the bar above denotes the Dirac adjoint. This equation controls the propagation of the wave function of composite objects such as the delta baryons or for the conjectural gravitino. So far, no elementary particle with spin 3/2 has been found experimentally. The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation , where is an arbitrary spinor field. This is simply the local supersymmetry of supergravity, and the field must be a gravitino. "Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
В теоретической физике уравнение Рарриты — Швингера является релятивистским уравнением поля для спиновых 3/2 фермионов в четырехмерном плоском пространстве-времени. Оно аналогично уравнению Дирака для фермионов со спином 1/2. Это уравнение было впервые введено Уильямом Раритой и Джулианом Швингером в 1941 году. В современной нотации его можно записать следующим образом:
In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation of spin 3/2 fermions in a four dimensional flat spacetime. It is similar to the Dirac equation for spin 1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941. In modern notation it can be written as:
where is the Levi Civita symbol,
are Dirac matrices (with ) and ,
is the mass,
,
and is a vector valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the representation of the Lorentz group, or rather, its part. This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:
where the bar above denotes the Dirac adjoint. This equation controls the propagation of the wave function of composite objects such as the delta baryons or for the conjectural gravitino. So far, no elementary particle with spin 3/2 has been found experimentally. The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation , where is an arbitrary spinor field. This is simply the local supersymmetry of supergravity, and the field must be a gravitino. "Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
где — символ Леви-Чивиты,
— матрицы Дирака (с и ),
— масса,
,
а — векторнозначный спинор с дополнительными компонентами по сравнению с четырехкомпонентным спинором в уравнении Дирака. Он соответствует представлению группы Лоренца, или, точнее, ее части. Это уравнение поля может быть получено как уравнение Эйлера — Лагранжа, соответствующее лагранжиану Рарриты — Швингера:
In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation of spin 3/2 fermions in a four dimensional flat spacetime. It is similar to the Dirac equation for spin 1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941. In modern notation it can be written as:
where is the Levi Civita symbol,
are Dirac matrices (with ) and ,
is the mass,
,
and is a vector valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the representation of the Lorentz group, or rather, its part. This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:
where the bar above denotes the Dirac adjoint. This equation controls the propagation of the wave function of composite objects such as the delta baryons or for the conjectural gravitino. So far, no elementary particle with spin 3/2 has been found experimentally. The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation , where is an arbitrary spinor field. This is simply the local supersymmetry of supergravity, and the field must be a gravitino. "Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
где черта над обозначает сопряженный по Дираку. Это уравнение описывает распространение волновой функции составных объектов, таких как дельта-барионы или гипотетического гравитино. На сегодняшний день экспериментально не обнаружено ни одной элементарной частицы со спином 3/2. Безмассовое уравнение Рарриты — Швингера обладает фермионной калибровочной симметрией: оно инвариантно относительно калибровочного преобразования , где — произвольное спинорное поле. Это просто локальная суперсимметрия супергравитации, и поле должно быть гравитино. Существуют также версии уравнения Рарриты — Швингера, соответствующие представлениям Вейля и Майораны.
In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation of spin 3/2 fermions in a four dimensional flat spacetime. It is similar to the Dirac equation for spin 1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941. In modern notation it can be written as:
where is the Levi Civita symbol,
are Dirac matrices (with ) and ,
is the mass,
,
and is a vector valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the representation of the Lorentz group, or rather, its part. This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:
where the bar above denotes the Dirac adjoint. This equation controls the propagation of the wave function of composite objects such as the delta baryons or for the conjectural gravitino. So far, no elementary particle with spin 3/2 has been found experimentally. The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation , where is an arbitrary spinor field. This is simply the local supersymmetry of supergravity, and the field must be a gravitino. "Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
Недостатки уравнения
Текущее описание массивных полей с высоким спином с использованием формализмов Рариты-Швингера или Фирца-Паули имеет ряд недостатков.