Dirac equation

The Dirac equation is a relativistic-invariant equation of motion for a bispinor classical field of an electron , also applicable to the description of other point fermions with spin 1/2; established by P. Dirac in 1928 .

Type of equation

The Dirac equation is written as

\left(mc^{2}\alpha _{0}+c\sum _{j=1}^{3}\alpha _{j}p_{j}\right)\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t),

{\ displaystyle \ left (mc ^ {2} \ alpha _ {0} + c \ sum _ {j = 1} ^ {3} \ alpha _ {j} p_ {j} \ right) \ psi (\ mathbf { x}, t) = i \ hbar {\ frac {\ partial \ psi} {\ partial t}} (\ mathbf {x}, t),}

\left(mc^{2}\alpha _{0}+c\sum _{j=1}^{3}\alpha _{j}p_{j}\right)\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t),

Where $m\$ ${\ displaystyle m \}$ $m\$ - mass of an electron (or other fermion described by the equation), $c\$ ${\ displaystyle c \}$ $c\$ Is the speed of light , $p_{j}=-i\hbar \partial _{j}$ ${\ displaystyle p_ {j} = - i \ hbar \ partial _ {j}}$ $p_j = - i \hbar \partial_j$ - three operators of momentum components (in x, y, z ), $\hbar ={h \over 2\pi }$ ${\ displaystyle \ hbar = {h \ over 2 \ pi}}$ $\hbar = {h \over 2 \pi}$ , $h$ ${\ displaystyle h}$ $h$ Is the Planck constant , x = ( x, y, z ) and t are the spatial coordinates and time, respectively, and $\psi (\mathbf {x} ,t)$ ${\ displaystyle \ psi (\ mathbf {x}, t)}$ $\psi(\mathbf{x},t)$ - four-component complex wave function (bispinor).

$\alpha _{0},\alpha _{1},\alpha _{2},\alpha _{3}\$ ${\ displaystyle \ alpha _ {0}, \ alpha _ {1}, \ alpha _ {2}, \ alpha _ {3} \}$ $\alpha _{0},\alpha _{1},\alpha _{2},\alpha _{3}\$ - linear operators over the bispinor space that act on the wave function ( Pauli matrices ). These operators are selected so that each pair of such operators anticommutes, and the square of each is equal to one:

\alpha _{i}\alpha _{j}=-\alpha _{j}\alpha _{i}\,

{\ displaystyle \ alpha _ {i} \ alpha _ {j} = - \ alpha _ {j} \ alpha _ {i} \,}

\alpha _{i}\alpha _{j}=-\alpha _{j}\alpha _{i}\,

Where

i\neq j

{\ displaystyle i \ neq j}

i\neq j

and indices

i,j\

{\ displaystyle i, j \}

i,j\

vary from 0 to 3,

\alpha _{i}^{2}=1

{\ displaystyle \ alpha _ {i} ^ {2} = 1}

\alpha _{i}^{2}=1

for

i\

{\ displaystyle i \}

i\

from 0 to 3.

In the presentation under discussion, these operators are represented by 4 × 4 matrices (this is the minimum size of matrices for which anti-commutation conditions are fulfilled), called Dirac alpha matrices

The entire operator in brackets on the left side of the equation is called the Dirac operator, more precisely, in modern terminology it should be called the Dirac Hamiltonian, since the Dirac operator is usually called the covariant operator D with which the Dirac equation is written as DΨ = 0 (as described in the following remark).

In modern physics, the covariant form of writing ^{[1] of} the Dirac equation is often used (see below for details):

\left(i\hbar c\,\gamma ^{\mu }\,\partial _{\mu }-mc^{2}\right)\psi =0.

{\ displaystyle \ left (i \ hbar c \, \ gamma ^ {\ mu} \, \ partial _ {\ mu} -mc ^ {2} \ right) \ psi = 0.}

\left(i\hbar c \, \gamma^\mu \, \partial_\mu - mc^2 \right) \psi = 0.

Physical meaning

Electron, positron

It follows from the Dirac equation that the electron has its own mechanical angular momentum - spin equal to ħ / 2, as well as an intrinsic magnetic moment equal to (without taking into account the gyromagnetic ratio) the Bohr magneton eħ / 2mc, which were previously discovered experimentally (1925) (e and m is the charge and mass of the electron, c is the speed of light, ħ is the Dirac constant (the reduced Planck constant)). Using the Dirac equation, a more accurate formula was obtained for the energy levels of a hydrogen atom (and hydrogen-like atoms), including a fine level structure (see Atom ), and the Zeeman effect was also explained. Based on the Dirac equation, formulas were found for the probabilities of scattering of photons by free electrons ( Compton effect ) and electron radiation during its deceleration ( bremsstrahlung ), which have received experimental confirmation. However, a consistent relativistic description of the motion of an electron is given by quantum electrodynamics .

A characteristic feature of the Dirac equation is the presence among its solutions of those that correspond to states with negative energy values for the free movement of the particle (which corresponds to the negative mass of the particle ). This represented a difficulty for the theory, since all mechanical laws for a particle in such states would be incorrect, and transitions to these states in quantum theory are possible. The actual physical meaning of transitions to levels with negative energy was clarified later on when the possibility of interconversion of particles was proved. From the Dirac equation it followed that there must be a new particle (antiparticle with respect to the electron) with the mass of the electron and an electric charge of the opposite sign; such a particle was indeed discovered in 1932 by K. Anderson and was called a positron . This was a huge success of the Dirac electron theory. The transition of an electron from a state with negative energy to a state with positive energy and the reverse transition are interpreted as the formation of an electron-positron pair and the annihilation of such a pair.

Application to other particles

The Dirac equation is valid not only for electrons, but also for other elementary particles with spin 1/2 (in units of ħ) - fermions , for example muons , neutrinos .

Moreover, good agreement with experiment is obtained by directly applying the Dirac equation to simple (rather than composite) particles, like those that have just been mentioned.

For a proton and a neutron (composite particles consisting of quarks bound by a gluon field , but also having 1/2 spin), when applied directly (as to simple particles), it leads to incorrect magnetic moments: the magnetic moment of the “Dirac” proton “should be »Is equal to the nuclear magneton eħ / 2Mc (M is the mass of the proton), and the neutron (since it is not charged) is zero. Experience shows that the magnetic moment of the proton is about 2.8 times greater than the nuclear magneton, and the magnetic moment of the neutron is negative and in absolute value is about 2/3 of the magnetic moment of the proton. This phenomenon is called the anomalous magnetic moment of the proton and neutron.

The anomalous magnetic moment of these particles indicates their internal structure, and is one of the important experimental confirmations of their quark structure.

In fact, this equation is applicable to quarks, which are also elementary particles with spin 1/2. The modified Dirac equation can be used to describe protons and neutrons , which are not elementary particles (they consist of quarks).

Dirac Equation and Quantum Field Theory

The Dirac equation does not describe the probability amplitude for a single electron, as it might seem, but the quantity associated with the charge density and current of a Dirac particle: due to the conservation of charge, a quantity that was considered the full probability of finding the particle is preserved. Thus, the Dirac equation is multiparticle from the very beginning.

A theory that includes only the Dirac equation, interacting with a classical external electromagnetic field, does not quite correctly take into account the birth and annihilation of particles. It well predicts the magnetic moment of the electron and the fine structure of the lines in the spectrum of atoms. It explains the electron spin, since two of the four solutions to the equation correspond to two electron spin states. The two remaining solutions with negative energy correspond to the electron antiparticle ( positron ) predicted by Dirac based on his theory and almost immediately afterwards discovered experimentally.

Despite these successes, such a theory has the disadvantage that it does not describe the interaction of a quantized electron field with a quantized electromagnetic field, including the birth / destruction of particles - one of the fundamental processes of the relativistic theory of interacting fields. This difficulty is resolved in quantum field theory . In the case of electrons, a quantized electromagnetic field is added, quantization of the electronic field itself and the interaction of these fields, and the resulting theory is called quantum electrodynamics .

Derivation of the Dirac equation

The Dirac equation is a relativistic generalization of the Schrödinger equation :

H\left|\psi (t)\right\rangle =i\hbar {d \over dt}\left|\psi (t)\right\rangle .

{\ displaystyle H \ left | \ psi (t) \ right \ rangle = i \ hbar {d \ over dt} \ left | \ psi (t) \ right \ rangle.}

For convenience, we will work in a coordinate representation in which the state of the system is defined by the wave function ψ ( x , t ). In this representation, the Schrödinger equation is written as

H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi (\mathbf {x} ,t)}{\partial t}},

{\ displaystyle H \ psi (\ mathbf {x}, t) = i \ hbar {\ frac {\ partial \ psi (\ mathbf {x}, t)} {\ partial t}},}

where the Hamiltonian H now acts on the wave function.

We must define the Hamiltonian so that it describes the total energy of the system. Consider a free electron (not interacting with anything, isolated from all extraneous fields). For the nonrelativistic model, we would take the Hamiltonian analogous to kinetic energy in classical mechanics (without taking into account either the relativistic corrections or the spin):

H=\sum _{j=1}^{3}{\frac {p_{j}^{2}}{2m}},

{\ displaystyle H = \ sum _ {j = 1} ^ {3} {\ frac {p_ {j} ^ {2}} {2m}},}

where p _j are the momentum projection operators, where the index j = 1,2,3 denotes the Cartesian coordinates. Each such operator acts on the wave function as a spatial derivative:

p_{j}\psi (\mathbf {x} ,t)\ {\stackrel {\mathrm {def} }{=}}\ -i\hbar \,{\frac {\partial \psi (\mathbf {x} ,t)}{\partial x_{j}}}.

{\ displaystyle p_ {j} \ psi (\ mathbf {x}, t) \ {\ stackrel {\ mathrm {def}} {=}} \ -i \ hbar \, {\ frac {\ partial \ psi (\ mathbf {x}, t)} {\ partial x_ {j}}}.

To describe a relativistic particle, we must find another Hamiltonian. In this case, there is reason to believe that the momentum operator saves the definition just given. According to the relativistic relation, the total energy of the system is expressed as

E={\sqrt {(mc^{2})^{2}+\sum _{j=1}^{3}(p_{j}c)^{2}}}.

{\ displaystyle E = {\ sqrt {(mc ^ {2}) ^ {2} + \ sum _ {j = 1} ^ {3} (p_ {j} c) ^ {2}}}.

This leads to the expression

{\sqrt {(mc^{2})^{2}+\sum _{j=1}^{3}(p_{j}c)^{2}}}\ \psi =i\hbar {\frac {d\psi }{dt}}.

{\ displaystyle {\ sqrt {(mc ^ {2}) ^ {2} + \ sum _ {j = 1} ^ {3} (p_ {j} c) ^ {2}}} \ \ psi = i \ hbar {\ frac {d \ psi} {dt}}.}

This is not a completely satisfactory equation, since there is no obvious Lorentz covariance (expressing the formal equality of time and spatial coordinates, which is one of the cornerstones of the special theory of relativity ), and in addition, the written root from the operator is not written out explicitly. However, squaring the left and right sides leads to the explicitly Lorentz-covariant Klein-Gordon equation . Dirac suggested that since the right-hand side of the equation contains the first time derivative, the left-hand side should have only first-order derivatives with respect to spatial coordinates (in other words, momentum operators in the first degree). Then, assuming that the coefficients of the derivatives, whatever their nature, are constants (due to the homogeneity of space), it remains only to write:

i\hbar {\frac {d\psi }{dt}}=\left[c\sum _{i=1}^{3}\alpha _{i}p_{i}+\alpha _{0}mc^{2}\right]\psi

{\ displaystyle i \ hbar {\ frac {d \ psi} {dt}} = \ left [c \ sum _ {i = 1} ^ {3} \ alpha _ {i} p_ {i} + \ alpha _ { 0} mc ^ {2} \ right] \ psi}

- this is the Dirac equation (for a free particle).

However, we have not yet determined the coefficients $\alpha _{i}\$ ${\ displaystyle \ alpha _ {i} \}$ . If the assumption of Dirac is correct, then the right side, squared, should give

(mc^{2})^{2}+\sum _{j=1}^{3}(p_{j}c)^{2},

{\ displaystyle (mc ^ {2}) ^ {2} + \ sum _ {j = 1} ^ {3} (p_ {j} c) ^ {2},}

i.e

\left(mc^{2}\alpha _{0}+c\sum _{j=1}^{3}\alpha _{j}p_{j}\,\right)^{2}=(mc^{2})^{2}+\sum _{j=1}^{3}(p_{j}c)^{2}.

{\ displaystyle \ left (mc ^ {2} \ alpha _ {0} + c \ sum _ {j = 1} ^ {3} \ alpha _ {j} p_ {j} \, \ right) ^ {2} = (mc ^ {2}) ^ {2} + \ sum _ {j = 1} ^ {3} (p_ {j} c) ^ {2}.}

Just opening the brackets on the left side of the resulting equation, we obtain the following conditions on α:

\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}=0\,,

{\ displaystyle \ alpha _ {i} \ alpha _ {j} + \ alpha _ {j} \ alpha _ {i} = 0 \ ,,}

for all

i,j=0,1,2,3(i\neq j),

{\ displaystyle i, j = 0,1,2,3 (i \ neq j),}

\alpha _{i}^{2}=1\,,

{\ displaystyle \ alpha _ {i} ^ {2} = 1 \ ,,}

for all

i=0,1,2,3.\ ,

{\ displaystyle i = 0,1,2,3. \,}

or, in short, writing everything together:

\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}=2\delta _{ij}\

{\ displaystyle \ alpha _ {i} \ alpha _ {j} + \ alpha _ {j} \ alpha _ {i} = 2 \ delta _ {ij} \}

for

\ i,j=0,1,2,3,

{\ displaystyle \ i, j = 0,1,2,3,}

or, even shorter, using curly brackets to denote anti-commutators:

\left\{\alpha _{i},\alpha _{j}\right\}=2\delta _{ij}\

{\ displaystyle \ left \ {\ alpha _ {i}, \ alpha _ {j} \ right \} = 2 \ delta _ {ij} \}

for

\ i,j=0,1,2,3.

{\ displaystyle \ i, j = 0,1,2,3.}

where {,} is the anticommutator defined as { A, B } ≡AB + BA , and δ _ij is the Kronecker symbol , which takes the value 1 if the two indices are equal and otherwise 0. See Clifford algebra .

Since such relations cannot be satisfied for ordinary numbers (because the numbers commute, but α does not), it remains - most simply - to assume that α are some linear operators or matrices (then the units and zeros on the right side of the relations can be considered unit and zero operator or matrix) and you can try to find a specific set of α using these relations (and this succeeds).

It is here that for the first time it becomes completely clear that the wave function should not be one-component (that is, not scalar), but vector, having in mind the vectors of some abstract “internal” space, not directly connected with ordinary physical space or space-time.

The matrices must be Hermitian , so that the Hamiltonian is also a Hermitian operator. The smallest dimension of matrices that satisfy the above criteria is complex 4 × 4 matrices, although their specific choice (or presentation ) is not unique. These matrices with the operation of matrix multiplication form a group. Although the choice of representation of this group does not affect the properties of the Dirac equation, it does affect the physical meaning of the components of the wave function. The wave function, obviously, should then be a four-dimensional complex abstract (not directly related to the ordinary space-time vectors) vector field (i.e., a bispinor field).

In the introduction, we presented the presentation used by Dirac. This view can be correctly written as

\alpha _{0}={\begin{bmatrix}I&0\\0&-I\end{bmatrix}},\quad \alpha _{j}={\begin{bmatrix}0&\sigma _{j}\\\sigma _{j}&0\end{bmatrix}},

{\ displaystyle \ alpha _ {0} = {\ begin {bmatrix} I & 0 \\ 0 & -I \ end {bmatrix}}, \ quad \ alpha _ {j} = {\ begin {bmatrix} 0 & \ sigma _ {j } \\\ sigma _ {j} & 0 \ end {bmatrix}},}

where 0 and I are 2 × 2 zero and identity matrices, respectively, and σ _j ( j = 1, 2, 3) are Pauli matrices , which, incidentally, are a matrix representation of quaternions , which have long been known that they anticommute.

Hamiltonian in this equation

H=\,mc^{2}\alpha _{0}+c\sum _{j=1}^{3}\alpha _{j}p_{j}

{\ displaystyle H = \, mc ^ {2} \ alpha _ {0} + c \ sum _ {j = 1} ^ {3} \ alpha _ {j} p_ {j}}

is called the Dirac Hamiltonian .

For the usual Dirac equation in two-dimensional space or in three-dimensional space, but with m = 0, instead of alpha matrices, just Pauli matrices are enough; instead of the four-component bispinor field, the role of the wave function will be played by the two-component spinor field.

The nature of the wave function

Since 4 × 4 matrices act on the wave function ψ , it must be a four-component object. We will see in the next section that the wave function consists of two degrees of freedom, one of which corresponds to positive energies and the other to negative ones. Each of them has two more degrees of freedom associated with the projection of the spin onto the selected direction, conditionally often denoted by the words “up” or “down”.

We can write the wave function in the form of a column:

\psi (\mathbf {x} ,t)\ {\stackrel {\mathrm {def} }{=}}\ {\begin{bmatrix}\psi _{1}(\mathbf {x} ,t)\\\psi _{2}(\mathbf {x} ,t)\\\psi _{3}(\mathbf {x} ,t)\\\psi _{4}(\mathbf {x} ,t)\end{bmatrix}}.

{\ displaystyle \ psi (\ mathbf {x}, t) \ {\ stackrel {\ mathrm {def}} {=}} \ {\ begin {bmatrix} \ psi _ {1} (\ mathbf {x}, t ) \\\ psi _ {2} (\ mathbf {x}, t) \\\ psi _ {3} (\ mathbf {x}, t) \\\ psi _ {4} (\ mathbf {x}, t) \ end {bmatrix}}.}

The dual wave function is written as a string:

{\bar {\psi }}\ {\stackrel {\mathrm {def} }{=}}{\bar {\psi }}(\mathbf {x} ,t)\ {\stackrel {\mathrm {def} }{=}}\ \psi ^{\dagger }\alpha ^{0},

{\ displaystyle {\ bar {\ psi}} \ {\ stackrel {\ mathrm {def}} {=}} {\ bar {\ psi}} (\ mathbf {x}, t) \ {\ stackrel {\ mathrm {def}} {=}} \ \ psi ^ {\ dagger} \ alpha ^ {0},}

Where

\psi ^{\dagger }={\begin{bmatrix}\psi _{1}^{*}(\mathbf {x} ,t)&\psi _{2}^{*}(\mathbf {x} ,t)&\psi _{3}^{*}(\mathbf {x} ,t)&\psi _{4}^{*}(\mathbf {x} ,t)\end{bmatrix}}

{\ displaystyle \ psi ^ {\ dagger} = {\ begin {bmatrix} \ psi _ {1} ^ {*} (\ mathbf {x}, t) & \ psi _ {2} ^ {*} (\ mathbf {x}, t) & \ psi _ {3} ^ {*} (\ mathbf {x}, t) & \ psi _ {4} ^ {*} (\ mathbf {x}, t) \ end {bmatrix }}}

the symbol * stands for ordinary complex conjugation .

As for the usual one-component wave function, we can introduce the squared modulus of the wave function, which gives the probability density as a function of the x coordinate and time t . In this case, the role of the square of the module is played by the scalar product of the wave function and its dual, that is, the square of the Hermitian norm of the bispinor:

{\bar {\psi }}\psi ={\bar {\psi }}(\mathbf {x} ,t)\psi \,(\mathbf {x} ,t)=\sum _{a,b=1}^{4}\psi _{a}^{*}(\mathbf {x} ,t)(\alpha ^{0})_{ab}\psi _{b}(\mathbf {x} ,t).

{\ displaystyle {\ bar {\ psi}} \ psi = {\ bar {\ psi}} (\ mathbf {x}, t) \ psi \, (\ mathbf {x}, t) = \ sum _ {a , b = 1} ^ {4} \ psi _ {a} ^ {*} (\ mathbf {x}, t) (\ alpha ^ {0}) _ {ab} \ psi _ {b} (\ mathbf { x}, t).}

Preservation of probability sets the normalization condition

\int {\bar {\psi }}\psi \;d^{3}x=1.

{\ displaystyle \ int {\ bar {\ psi}} \ psi \; d ^ {3} x = 1.}

Using the Dirac equation, we can obtain a “local” probability current :

{\frac {\partial }{\partial t}}{\bar {\psi }}\psi \,(\mathbf {x} ,t)=-\nabla \cdot \mathbf {J} .

{\ displaystyle {\ frac {\ partial} {\ partial t}} {\ bar {\ psi}} \ psi \, (\ mathbf {x}, t) = - \ nabla \ cdot \ mathbf {J}.}

The probability current J is defined as

J_{j}=c{\bar {\psi }}\alpha _{j}\psi .

{\ displaystyle J_ {j} = c {\ bar {\ psi}} \ alpha _ {j} \ psi.}

Multiplying J by the electron charge e , we arrive at the electric current density j for the electron.

The value of the components of the wave function depends on the coordinate system. Dirac showed how ψ transforms when the coordinate system changes, including rotations in three-dimensional space and transformations between (quickly) reference frames moving relative to each other. ψ in this case does not transform as a vector of ordinary space (or space-time) during space rotations or Lorentz transformations (что само по себе и не удивительно, так как его компоненты изначально не связаны прямо с направлениями в обычном пространстве). Такой объект получил название четырёхкомпонентного дираковского спинора (иначе называемого биспинором — последнее название связано с тем, что первоначально в качестве спиноров рассматривались только двухкомпонентные комплексные объекты, пара которых может образовать биспинор). Биспинор можно интерпретировать как вектор в особом пространстве, называемом обычно «внутренним пространством», не пересекающемся с обычным («внешним») пространством. Однако, как уже было сказано выше, компоненты спинорных волновых функций при преобразовании координат внешнего пространства изменяются вполне определённым образом, хотя и отличающемся от преобразования компонент векторов обычного пространства.

Точности ради следует сказать, что все изменения, связанные с поворотами координат во внешнем пространстве, можно перенести на матрицы α (которые тогда будут выглядеть по-разному для разных внешних систем координат, но будут сохранять свои основные свойства — антикоммутативности и равенства единице квадрата каждой матрицы), в этом случае компоненты (би-)спиноров вообще не будут меняться при поворотах внешнего пространства.

Решение уравнения

Для решения уравнения в случае свободной частицы привлекается спинор $\chi$ ${\ displaystyle \ chi}$

\chi ^{(1)}={\begin{bmatrix}1\\0\end{bmatrix}},\quad \quad \chi ^{(2)}={\begin{bmatrix}0\\1\end{bmatrix}},

\chi ^{(1)}={\begin{bmatrix}1\\0\end{bmatrix}},\quad \quad \chi ^{(2)}={\begin{bmatrix}0\\1\end{bmatrix}},

Where $\chi ^{(1)}$ $\chi ^{(1)}$ соответствует спину вверх , а $\chi ^{(2)}$ $\chi ^{(2)}$ соответствует спину вниз .

Для античастиц верно обратное:

\chi ^{*(1)}={\begin{bmatrix}0\\1\end{bmatrix}},\quad \quad \chi ^{*(2)}={\begin{bmatrix}1\\0\end{bmatrix}}.

\chi ^{*(1)}={\begin{bmatrix}0\\1\end{bmatrix}},\quad \quad \chi ^{*(2)}={\begin{bmatrix}1\\0\end{bmatrix}}.

Введём также матрицы Паули ,

\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.

\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.

Для частиц

Решение уравнения Дирака для свободных частиц запишется в виде

\psi =u(\mathbf {p} )e^{ip\cdot x},

\psi =u(\mathbf {p} )e^{ip\cdot x},

Where

\mathbf {p}

\mathbf {p}

— обычный трёхмерный вектор, а

p и x — 4-векторы .

Биспинор u является функцией момента и спина,

u^{(s)}(\mathbf {p} )={\sqrt {E+m}}{\begin{bmatrix}\chi ^{(s)}\\{\frac {\mathbf {\sigma } \cdot \mathbf {p} }{E+m}}\chi ^{(s)}\end{bmatrix}}.

u^{(s)}(\mathbf {p} )={\sqrt {E+m}}{\begin{bmatrix}\chi ^{(s)}\\{\frac {\mathbf {\sigma } \cdot \mathbf {p} }{E+m}}\chi ^{(s)}\end{bmatrix}}.

Для античастиц

\psi =v(\mathbf {p} )e^{ip\cdot x}

\psi =v(\mathbf {p} )e^{ip\cdot x}

with

v^{(s)}(\mathbf {p} )={\sqrt {|E|+m}}{\begin{bmatrix}{\frac {-\mathbf {\sigma } \cdot \mathbf {p} }{|E|+m}}\chi ^{*(s)}\\\chi ^{*(s)}\end{bmatrix}}.

v^{(s)}(\mathbf {p} )={\sqrt {|E|+m}}{\begin{bmatrix}{\frac {-\mathbf {\sigma } \cdot \mathbf {p} }{|E|+m}}\chi ^{*(s)}\\\chi ^{*(s)}\end{bmatrix}}.

Биспиноры

Соотношения полноты для биспиноров u и v :

\sum _{s=1,2}{u_{p}^{(s)}{\bar {u}}_{p}^{(s)}}=p\!\!\!/+m,

\sum _{s=1,2}{u_{p}^{(s)}{\bar {u}}_{p}^{(s)}}=p\!\!\!/+m,

\sum _{s=1,2}{v_{p}^{(s)}{\bar {v}}_{p}^{(s)}}=p\!\!\!/-m,

\sum _{s=1,2}{v_{p}^{(s)}{\bar {v}}_{p}^{(s)}}=p\!\!\!/-m,

Where

p\!\!\!/=\gamma ^{\mu }p_{\mu }

p\!\!\!/=\gamma ^{\mu }p_{\mu }

(определение

\gamma ^{\mu }

\gamma ^{\mu }

— см. чуть ниже).

Энергетический спектр

Полезно найти собственные значения энергии гамильтониана Дирака. Для того чтобы это сделать, мы должны решить стационарное уравнение:

H\psi _{0}(\mathbf {x} )=E\psi _{0}(\mathbf {x} ),

H\psi _{0}(\mathbf {x} )=E\psi _{0}(\mathbf {x} ),

где ψ ₀ — независимая от времени часть полной волновой функции

\psi (\mathbf {x} ,t)=\psi _{0}(\mathbf {x} )e^{-iEt/\hbar },

\psi (\mathbf {x} ,t)=\psi _{0}(\mathbf {x} )e^{-iEt/\hbar },

подстановкой которой в нестационарное уравнение Дирака мы получаем стационарное.

Будем искать решение в виде плоских волн. Для удобства выберем в качестве оси движения ось z . Таким образом

\psi _{0}=we^{\frac {ipz}{\hbar }},

\psi _{0}=we^{\frac {ipz}{\hbar }},

где w — постоянный четырёхкомпонентный спинор и p — импульс частицы, как можно показать действуя оператором импульса на эту волновую функцию. В представлении Дирака уравнение для ψ ₀ сводится к задаче на собственные значения:

{\begin{bmatrix}mc^{2}&0&pc&0\\0&mc^{2}&0&-pc\\pc&0&-mc^{2}&0\\0&-pc&0&-mc^{2}\end{bmatrix}}w=Ew.

{\begin{bmatrix}mc^{2}&0&pc&0\\0&mc^{2}&0&-pc\\pc&0&-mc^{2}&0\\0&-pc&0&-mc^{2}\end{bmatrix}}w=Ew.

Для каждого значения p , существует два двумерных пространства собственных значений. Одно пространство собственных значений содержит положительные собственные значения, а другое — отрицательные в виде

E_{\pm }(p)=\pm {\sqrt {(mc^{2})^{2}+(pc)^{2}}}.

E_{\pm }(p)=\pm {\sqrt {(mc^{2})^{2}+(pc)^{2}}}.

пространство с положительными собственными значениями порождается собственными состояниями:

\left\{{\begin{bmatrix}pc\\0\\\epsilon \\0\end{bmatrix}}\,,\,{\begin{bmatrix}0\\pc\\0\\-\epsilon \end{bmatrix}}\right\}\times {\frac {1}{\sqrt {\epsilon ^{2}+(pc)^{2}}}}

\left\{{\begin{bmatrix}pc\\0\\\epsilon \\0\end{bmatrix}}\,,\,{\begin{bmatrix}0\\pc\\0\\-\epsilon \end{bmatrix}}\right\}\times {\frac {1}{\sqrt {\epsilon ^{2}+(pc)^{2}}}}

и для отрицательных:

\left\{{\begin{bmatrix}-\epsilon \\0\\pc\\0\end{bmatrix}}\,,\,{\begin{bmatrix}0\\\epsilon \\0\\pc\end{bmatrix}}\right\}\times {\frac {1}{\sqrt {\epsilon ^{2}+(pc)^{2}}}},

\left\{{\begin{bmatrix}-\epsilon \\0\\pc\\0\end{bmatrix}}\,,\,{\begin{bmatrix}0\\\epsilon \\0\\pc\end{bmatrix}}\right\}\times {\frac {1}{\sqrt {\epsilon ^{2}+(pc)^{2}}}},

Where

\epsilon \ {\stackrel {\mathrm {def} }{=}}\ |E|-mc^{2}.

\epsilon \ {\stackrel {\mathrm {def} }{=}}\ |E|-mc^{2}.

Первое порождающее собственное состояние в каждом собственном пространстве имеет положительную проекцию спина на z направление («спин вверх»), и второе собственное состояние имеет спин указывающий в противоположном направлении − z («спин вниз»).

В нерелятивистском пределе ε компонента спинора уменьшается до кинетической энергии частицы, которая пренебрежимо мала в сравнении с pc :

\epsilon \sim {\frac {p^{2}}{2m}}\ll pc.

\epsilon \sim {\frac {p^{2}}{2m}}\ll pc.

В этом пределе четырёхкомпонентную волновую функцию можно интерпретировать как относительную амплитуду (i) спин вверх с положительной энергией, (ii) спин вниз с положительной энергией, (iii) спин вверх с отрицательной энергией, и (iv) спин вниз с отрицательной энергией. Это описание не точно в релятивистском случае, где ненулевые компоненты спинора имеют тот же порядок величины.

Дырочная теория

The solutions with negative energies found in the previous section are problematic, since it was assumed that the particle has positive energy. Mathematically speaking, however, there seems to be no reason for us to reject negative energy solutions. Since they exist, we cannot just ignore them, as soon as we turn on the interaction between the electron and the electromagnetic field, any electron placed in a state with positive energy would go into a state with negative energy, successfully lowering the energy, emitting excess energy in the form of photons . Real electrons obviously do not behave this way.

To deal with this problem, Dirac introduced the hypothesis known as the hole theory that vacuum is a many-particle quantum state in which all states with negative energy are occupied. This description of the vacuum as the "sea" of electrons is called the Dirac Sea . Since the Pauli Inhibition Principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a state with positive energy, and electrons with positive energy would not go into states with negative energy.

Dirac further argued that if states with negative energy were not completely filled, each unoccupied state - called a hole - would behave like a positively charged particle. The hole has “positive” energy, since energy is needed to create a particle-hole pair from a vacuum. As noted above, Dirac initially thought that the hole could be a proton, but Weil indicated that the hole should behave as if it has the same mass as an electron, while the proton is more than 1800 times heavier. The hole was ultimately identified as a positron experimentally discovered by Karl Anderson in 1932 .

The description of the "vacuum" through an infinite sea of negative energy electrons is not entirely satisfactory. The infinitely negative contributions from a sea of negative energy electrons should be reduced with infinite positive “bare” energy and a contribution to the charge density, and the current coming from the sea of negative energy electrons will precisely decrease with an infinite positive “jelly” background so that the total electric charge density of the vacuum equal to zero. In quantum field theory , Bogolyubov’s transformation of the creation and annihilation operators (transforming an occupied electronic state with negative energy into an empty positron state with positive energy and an empty electronic state with negative energy into an occupied positron state with positive energy) allows us to bypass the Dirac Sea formalism even though , which, formally, these approaches are equivalent.

In certain applications in solid state physics , however, the basic concepts of “hole theory” are correct. The sea of conduction electrons in a conductor, called the Fermi sea , contains electrons with energies up to the chemical potential of the system. Unfilled states in the Fermi sea behave like positively charged electrons, although this is a “hole” and not a “positron”. The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.

Dirac equation in the representation of quaternions

The Dirac equation can simply be written in a representation using quaternions . We will write it in terms of representing two fields above the quaternions for the right (Ψ) and left (Φ) electrons:

\partial _{t}\psi i+i\partial _{x}\psi +j\partial _{y}\psi +k\partial _{z}\psi =m_{e}\phi j,

{\ displaystyle \ partial _ {t} \ psi i + i \ partial _ {x} \ psi + j \ partial _ {y} \ psi + k \ partial _ {z} \ psi = m_ {e} \ phi j ,}

\partial _{t}\phi i-i\partial _{x}\phi -j\partial _{y}\phi -k\partial _{z}\phi =m_{e}\psi j.

{\ displaystyle \ partial _ {t} \ phi ii \ partial _ {x} \ phi -j \ partial _ {y} \ phi -k \ partial _ {z} \ phi = m_ {e} \ psi j.}

It is important on which side single quaternions multiply. Note that the mass and time terms are multiplied on the right by quaternions. This representation of the Dirac equation is used in computer simulation.

Relativistically covariant form

The covariant notation for the Dirac equation for a free particle looks like this:

\left(i\hbar c\,\sum _{\mu =0}^{3}\;\gamma ^{\mu }\,\partial _{\mu }-mc^{2}\right)\psi =0,

{\ displaystyle \ left (i \ hbar c \, \ sum _ {\ mu = 0} ^ {3} \; \ gamma ^ {\ mu} \, \ partial _ {\ mu} -mc ^ {2} \ right) \ psi = 0,}

or, using the Einstein rule of summation over a repeating index, like this:

\left(i\hbar c\,\gamma ^{\mu }\,\partial _{\mu }-mc^{2}\right)\psi =0.

{\ displaystyle \ left (i \ hbar c \, \ gamma ^ {\ mu} \, \ partial _ {\ mu} -mc ^ {2} \ right) \ psi = 0.}

Explanation

It is often useful to use the Dirac equation in a relativistically covariant form in which spatial and temporal coordinates are considered formally on an equal footing.

To do this, first recall that the momentum operator p acts as a spatial derivative:

\mathbf {p} \psi (\mathbf {x} ,t)=-i\hbar \nabla \psi (\mathbf {x} ,t).

{\ displaystyle \ mathbf {p} \ psi (\ mathbf {x}, t) = - i \ hbar \ nabla \ psi (\ mathbf {x}, t).}

Multiplying the Dirac equation on each side by α ₀ (remembering that α ₀ ² = I ) and substituting it in the definition for p , we obtain

\left[i\hbar c\left(\alpha _{0}{\frac {\partial }{c\partial t}}+\sum _{j=1}^{3}\alpha _{0}\alpha _{j}{\frac {\partial }{\partial x_{j}}}\right)-mc^{2}\right]\psi =0.

{\ displaystyle \ left [i \ hbar c \ left (\ alpha _ {0} {\ frac {\ partial} {c \ partial t}} + \ sum _ {j = 1} ^ {3} \ alpha _ { 0} \ alpha _ {j} {\ frac {\ partial} {\ partial x_ {j}}} \ right) -mc ^ {2} \ right] \ psi = 0.}

Now we define four gamma matrices :

\gamma ^{0}\ {\stackrel {\mathrm {def} }{=}}\ \alpha _{0}\,,\quad \gamma ^{j}\ {\stackrel {\mathrm {def} }{=}}\ \alpha _{0}\alpha _{j}.

{\ displaystyle \ gamma ^ {0} \ {\ stackrel {\ mathrm {def}} {=}} \ \ alpha _ {0} \ ,, \ quad \ gamma ^ {j} \ {\ stackrel {\ mathrm { def}} {=}} \ \ alpha _ {0} \ alpha _ {j}.}

These matrices have the property that

\left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }\cdot I\,,\quad \mu ,\nu =0,1,2,3,

{\ displaystyle \ left \ {\ gamma ^ {\ mu}, \ gamma ^ {\ nu} \ right \} = 2 \ eta ^ {\ mu \ nu} \ cdot I \ ,, \ quad \ mu, \ nu = 0,1,2,3,}

where η is the metric of flat space. These relationships define a Clifford algebra called the Dirac algebra .

The Dirac equation can now be written using the four-vector x = ( ct , x ), as

\left(i\hbar c\,\sum _{\mu =0}^{3}\;\gamma ^{\mu }\,\partial _{\mu }-mc^{2}\right)\psi =0.

{\ displaystyle \ left (i \ hbar c \, \ sum _ {\ mu = 0} ^ {3} \; \ gamma ^ {\ mu} \, \ partial _ {\ mu} -mc ^ {2} \ right) \ psi = 0.}

In this form, the Dirac equation can be obtained by finding the extremum of the action

{\mathcal {S}}=\int {\bar {\psi }}(i\hbar c\,\sum _{\mu }\gamma ^{\mu }\partial _{\mu }-mc^{2})\psi \,d^{4}x,

{\ displaystyle {\ mathcal {S}} = \ int {\ bar {\ psi}} (i \ hbar c \, \ sum _ {\ mu} \ gamma ^ {\ mu} \ partial _ {\ mu} - mc ^ {2}) \ psi \, d ^ {4} x,}

Where

{\bar {\psi }}\ {\stackrel {\mathrm {def} }{=}}\ \psi ^{\dagger }\gamma _{0}

{\ displaystyle {\ bar {\ psi}} \ {\ stackrel {\ mathrm {def}} {=}} \ \ psi ^ {\ dagger} \ gamma _ {0}}

is called the Dirac adjoint matrix for ψ . This is the basis for using the Dirac equation in quantum field theory .

In this form, electromagnetic interaction can simply be added by expanding the partial derivative to a gauge-covariant derivative :

\partial _{\mu }\rightarrow D_{\mu }=\partial _{\mu }-ieA_{\mu }.

{\ displaystyle \ partial _ {\ mu} \ rightarrow D _ {\ mu} = \ partial _ {\ mu} -ieA _ {\ mu}.}

Recording Using Feynman Slash

Sometimes recording using “crossed matrices” (“Feynman slash”) is used. Having accepted the designation

a\!\!\!/\leftrightarrow \sum _{\mu }\gamma ^{\mu }a_{\mu },

{\ displaystyle a \! \! \!! / \ leftrightarrow \ sum _ {\ mu} \ gamma ^ {\ mu} a _ {\ mu},}

we see that the Dirac equation can be written as

(i\hbar c\,\partial \!\!\!/-mc^{2})\psi =0

{\ displaystyle (i \ hbar c \, \ partial \! \!!!!! - - mc ^ {2}) \ psi = 0}

and the expression for the action is written as

{\mathcal {S}}=\int {\bar {\psi }}(i\hbar c\,\partial \!\!\!/-mc^{2})\psi \,d^{4}x.

{\ displaystyle {\ mathcal {S}} = \ int {\ bar {\ psi}} (i \ hbar c \, \ partial \! \! \! / - mc ^ {2}) \ psi \, d ^ {4} x.}

Dirac bilinear forms

There are five different (neutral) Dirac bilinear forms without derivatives:

(S) scalar : ${\bar {\psi }}\psi$ ${\ displaystyle {\ bar {\ psi}} \ psi}$ (scalar, P-even)
(P) pseudoscalar : ${\bar {\psi }}\gamma ^{5}\psi$ ${\ displaystyle {\ bar {\ psi}} \ gamma ^ {5} \ psi}$ (scalar, P-odd)
(V) vector : ${\bar {\psi }}\gamma ^{\mu }\psi$ ${\ displaystyle {\ bar {\ psi}} \ gamma ^ {\ mu} \ psi}$ (vector, P-even)
(A) axial vector : ${\bar {\psi }}\gamma ^{\mu }\gamma ^{5}\psi$ ${\ displaystyle {\ bar {\ psi}} \ gamma ^ {\ mu} \ gamma ^ {5} \ psi}$ (vector, P-odd)
(T) tensor : ${\bar {\psi }}\sigma ^{\mu \nu }\psi$ ${\ displaystyle {\ bar {\ psi}} \ sigma ^ {\ mu \ nu} \ psi}$ (antisymmetric tensor)

Where $\sigma ^{\mu \nu }={\frac {i}{2}}\left[\gamma ^{\mu },\gamma ^{\nu }\right]_{-}$ ${\ displaystyle \ sigma ^ {\ mu \ nu} = {\ frac {i} {2}} \ left [\ gamma ^ {\ mu}, \ gamma ^ {\ nu} \ right] _ {-}}$ and $\gamma ^{5}=\gamma _{5}={\frac {i}{4!}}\epsilon _{\mu \nu \rho \lambda }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\lambda }=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}$ ${\ displaystyle \ gamma ^ {5} = \ gamma _ {5} = {\ frac {i} {4!}} \ epsilon _ {\ mu \ nu \ rho \ lambda} \ gamma ^ {\ mu} \ gamma ^ {\ nu} \ gamma ^ {\ rho} \ gamma ^ {\ lambda} = i \ gamma ^ {0} \ gamma ^ {1} \ gamma ^ {2} \ gamma ^ {3}}$ .

Electromagnetic Interaction

So far, we have considered an electron that is not affected by any external fields. By analogy with the Hamiltonian of a charged particle in classical electrodynamics , we can change the Dirac Hamiltonian so as to include the effect of the electromagnetic field . Rewritten Hamiltonian - (in SI units ):

H=\alpha _{0}mc^{2}+\sum _{j=1}^{3}\alpha _{j}\left[p_{j}-eA_{j}(\mathbf {x} ,t)\right]c+e\varphi (\mathbf {x} ,t),

{\ displaystyle H = \ alpha _ {0} mc ^ {2} + \ sum _ {j = 1} ^ {3} \ alpha _ {j} \ left [p_ {j} -eA_ {j} (\ mathbf {x}, t) \ right] c + e \ varphi (\ mathbf {x}, t),}

where e is the electric charge of the electron (it is agreed here that the sign of e is negative), and A and φ are the electromagnetic vector and scalar potentials, respectively.

Putting φ = 0 and working in the nonrelativistic limit, Dirac found wave functions for the two upper components in the positive energy region (which, as discussed earlier, are the dominant components in the nonrelativistic limit):

\left({\frac {1}{2m}}\sum _{j}|p_{j}-eA_{j}(\mathbf {x} ,t)|^{2}-{\frac {\hbar e}{2mc}}\sum _{j}\sigma _{j}B_{j}(\mathbf {x} )\right){\begin{bmatrix}\psi _{1}\\\psi _{2}\end{bmatrix}}

{\ displaystyle \ left ({\ frac {1} {2m}} \ sum _ {j} | p_ {j} -eA_ {j} (\ mathbf {x}, t) | ^ {2} - {\ frac {\ hbar e} {2mc}} \ sum _ {j} \ sigma _ {j} B_ {j} (\ mathbf {x}) \ right) {\ begin {bmatrix} \ psi _ {1} \\\ psi _ {2} \ end {bmatrix}}}

=(E-mc^{2}){\begin{bmatrix}\psi _{1}\\\psi _{2}\end{bmatrix}},

{\ displaystyle = (E-mc ^ {2}) {\ begin {bmatrix} \ psi _ {1} \\\ psi _ {2} \ end {bmatrix}},}

where B = $\nabla$ ${\ displaystyle \ nabla}$ × A is the magnetic field acting on the particle. This is the Pauli equation for nonrelativistic particles with a half-integer spin, with a magnetic moment $\hbar e/2mc$ ${\ displaystyle \ hbar e / 2mc}$ (that is, the g-factor equals 2). The actual magnetic moment of the electron is greater than this value, although only by about 0.12%. The mismatch occurs due to neglected quantum oscillations in the electromagnetic field. See vertex function .

For several years after the discovery of the Dirac equation, most physicists believed that it also describes a proton and a neutron , which are fermions with a half-integer spin. However, starting with the Stern and Frisch experiments in 1933 , the magnetic moments found for these particles do not coincide significantly with the values predicted from the Dirac equation. The proton has a magnetic moment 2.79 times greater than predicted (with the proton mass inserted for m in the above formulas), that is, the g factor is 5.58. A neutron that is electrically neutral has a g factor of −3.83. These “abnormal magnetic moments” were the first experimental sign that the proton and neutron are not elementary (but compound or, more generally, having some internal structure) particles. Subsequently, it turned out that they can be considered to consist of smaller particles called quarks , which are believed to be connected by a gluon field . Quarks have a half-integer spin and, as far as is known, are precisely described by the Dirac equation.

Hamiltonian of interaction

It is noteworthy that the Hamiltonian can be written as the sum of two terms:

H=H_{\mathrm {free} }+H_{\mathrm {int} },

{\ displaystyle H = H _ {\ mathrm {free}} + H _ {\ mathrm {int}},}

where H _free is the Dirac Hamiltonian for a free electron and H _int is the interaction Hamiltonian of an electron with an electromagnetic field. The last is written as

H_{\mathrm {int} }=e\varphi (\mathbf {x} ,t)-ec\sum _{j=1}^{3}\alpha _{j}A_{j}(\mathbf {x} ,t).

{\ displaystyle H _ {\ mathrm {int}} = e \ varphi (\ mathbf {x}, t) -ec \ sum _ {j = 1} ^ {3} \ alpha _ {j} A_ {j} (\ mathbf {x}, t).}

It has a mathematical expectation (average)

\langle H_{\mathrm {int} }\rangle =\int \,\psi ^{\dagger }H_{\mathrm {int} }\psi \,d^{3}x=\int \,\left(\rho \varphi -\sum _{i=1}^{3}j_{i}A_{i}\right)\,d^{3}x,

{\ displaystyle \ langle H _ {\ mathrm {int}} \ rangle = \ int \, \ psi ^ {\ dagger} H _ {\ mathrm {int}} \ psi \, d ^ {3} x = \ int \, \ left (\ rho \ varphi - \ sum _ {i = 1} ^ {3} j_ {i} A_ {i} \ right) \, d ^ {3} x,}

where ρ is the electric charge density and j is the electric current density defined through ψ . The integrand in the last integral - the interaction energy density - is a Lorentz-invariant scalar quantity, which is easy to see by writing in terms of the four-dimensional current density j = ( ρc , j ) and the four-dimensional electromagnetic potential A = ( φ / c , A ) - each of which is a 4-vector , and hence their scalar product is invariant. And the interaction energy is written as an integral over space from this invariant:

\langle H_{\mathrm {int} }\rangle =\int \,\left(\sum _{\mu ,\nu =0}^{3}\eta ^{\mu \nu }j_{\mu }A_{\nu }\right)\;d^{3}x,

{\ displaystyle \ langle H _ {\ mathrm {int}} \ rangle = \ int \, \ left (\ sum _ {\ mu, \ nu = 0} ^ {3} \ eta ^ {\ mu \ nu} j_ { \ mu} A _ {\ nu} \ right) \; d ^ {3} x,}

where η is the metric of a flat Minkowski space (Lorentzian space-time metric):

\eta ^{00}=1,\

{\ displaystyle \ eta ^ {00} = 1, \}

\eta ^{ii}\;=-1\quad \,(i=1,2,3),

{\ displaystyle \ eta ^ {ii} \; = - 1 \ quad \, (i = 1,2,3),}

\eta ^{\mu \nu }=0\ \ \ \ (\mu ,\nu =0,1,2,3;\mu \neq \nu ).

{\ displaystyle \ eta ^ {\ mu \ nu} = 0 \ \ \ \ (\ mu, \ nu = 0,1,2,3; \ mu \ neq \ nu).}

Consequently, the time-integrated energy interaction will give a Lorentz-invariant term in action (since the Lorentz rotations and transformations do not change the four-dimensional volume).

Lagrangian

The classical density of the fermion Lagrangian with a half-integer spin with mass m is given by

{\mathcal {L}}={\overline {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi ,

{\ displaystyle {\ mathcal {L}} = {\ overline {\ psi}} \ left (i \ gamma ^ {\ mu} \ partial _ {\ mu} -m \ right) \ psi,}

Where ${\overline {\psi }}=\psi ^{\dagger }\gamma ^{0}.$ ${\ displaystyle {\ overline {\ psi}} = \ psi ^ {\ dagger} \ gamma ^ {0}.}$

To obtain the equations of motion, we can substitute this Lagrangian into the Euler - Lagrange equations :

\partial _{\mu }\left({\frac {\partial L}{\partial (\partial _{\mu }\psi _{\sigma })}}\right)-{\frac {\partial L}{\partial \psi _{\sigma }}}=0.

{\ displaystyle \ partial _ {\ mu} \ left ({\ frac {\ partial L} {\ partial (\ partial _ {\ mu} \ psi _ {\ sigma})}} right) - {\ frac { \ partial L} {\ partial \ psi _ {\ sigma}}} = 0.}

Having rated two members:

{\frac {\partial L}{\partial (\partial _{\mu }\psi _{\sigma })}}={\overline {\psi }}_{\sigma ^{\prime }}\left(i\gamma ^{\mu }\right)_{\sigma ^{\prime }\sigma },

{\ displaystyle {\ frac {\ partial L} {\ partial (\ partial _ {\ mu} \ psi _ {\ sigma})}} = {\ overline {\ psi}} _ {\ sigma ^ {\ prime} } \ left (i \ gamma ^ {\ mu} \ right) _ {\ sigma ^ {\ prime} \ sigma},}

{\frac {\partial L}{\partial \psi _{\sigma }}}=-m{\overline {\psi }}_{\sigma },

{\ displaystyle {\ frac {\ partial L} {\ partial \ psi _ {\ sigma}}} = - m {\ overline {\ psi}} _ {\ sigma},}

And collecting both results, we get the equation

i\partial _{\mu }{\overline {\psi }}\gamma ^{\mu }+m{\overline {\psi }}=0,

{\ displaystyle i \ partial _ {\ mu} {\ overline {\ psi}} \ gamma ^ {\ mu} + m {\ overline {\ psi}} = 0,}

which is identical to the Dirac equation :

i\gamma ^{\mu }\partial _{\mu }\psi -m\psi =0.

{\ displaystyle i \ gamma ^ {\ mu} \ partial _ {\ mu} \ psi -m \ psi = 0.}

Notes

↑ Since the form with alpha matrices is Lorentz covariant, it is more correct to call the form with gamma matrices simply four-dimensional (and when replacing ordinary derivatives with covariant ones, it will give a general covariant notation of the Dirac equation)

Literature

Björken J. D. , Drell S. D. Relativistic quantum theory. - M .: Nauka, 1978. - T. 1. - 296 p.
Dyson F. Relativistic quantum mechanics. - Izhevsk: RHD, 2009 .-- 248 p.
Dirac P. A. M. Principles of quantum mechanics. - M .: Nauka, 1979.- 440 p.
Dirac P. A. M. Relativistic wave equation of an electron (Russian) // Uspekhi Fizicheskikh Nauk. - 1979. - T. 129 , no. 4 . - S. 681-691 .
Zee E. Quantum Field Theory in a Nutshell. - Izhevsk: RHD, 2009 .-- 632 p.
Peskin M., Schroeder D. Introduction to quantum field theory. - Izhevsk: RHD, 2001 .-- 784 p.
Schiff L. Quantum Mechanics. - M .: IL, 1959.- 476 p.
Shankar R. Principles of Quantum Mechanics. - Plenum, 1994.
Thaller B. The Dirac Equation. - Springer, 1992.
Dirac equation in the "Physical Encyclopedia"

Featured Articles

PAM Dirac "The Quantum Theory of the Electron", Proc. R. Soc. A117 610 (1928) , doi: http://doi.org/10.1098/rspa.1928.0023 - link to the volume of the Proceedings of the Royal Society of London containing the article at page 610
PAM Dirac "A Theory of Electrons and Protons", Proc. R. Soc. A126 360 (1930) link to the volume of the Proceedings of the Royal Society of London containing the article at page 360
CD Anderson, Phys. Rev. 43 , 491 (1933)
R. Frisch and O. Stern, Z. Phys. 85 4 (1933)

Links

Lectures on quantum physics

[1] Since the form with alpha matrices is Lorentz covariant, it is more correct to call the form with gamma matrices simply four-dimensional (and when replacing ordinary derivatives with covariant ones, it will give a general covariant notation of the Dirac equation)

Dirac equation

Type of equation

Physical meaning

Electron, positron

Application to other particles

Dirac Equation and Quantum Field Theory

Derivation of the Dirac equation

The nature of the wave function

Решение уравнения

Для частиц

Для античастиц

Биспиноры

Энергетический спектр

Дырочная теория

Dirac equation in the representation of quaternions

Relativistically covariant form

Explanation

Recording Using Feynman Slash

Dirac bilinear forms

Electromagnetic Interaction

Hamiltonian of interaction

Lagrangian

See also

Notes

Literature

Featured Articles

Links

More articles: