MECÂNICA GRACELI GENENRALIZADA - quântica TENSORIAL DIMENSIONAL RELATTIVISTA DE CAMPOS

Einstein field equations: describe interaction of matter with the gravitational field (massless spin-2 field):
$R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }\,R+g_{\mu \nu }\Lambda ={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$
The solution is a metric tensor field, rather than a wave function.

MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE CAMPOS.

MECÃNICA GRACELI GERAL - QTDRC.

$equação Graceli dimensional relativista tensorial quântica de campos$

$\psi ^{*}$

$[$ $\psi ^{*}$ $\psi ^{*}$ $G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $T_{\mu \nu }$ $/$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ /

$G_{\mu \nu }\$ $T_{\mu \nu }$ G $T^{\nu }{}_{\alpha }$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ $.$ = $\left|\psi ({\vec {r}},t)\right\rangle$ +

$\psi ^{*}$

//////

[

$\psi ^{*}$

$TeoriaInteraçãomediadorMagnitude relativaComportamentoFaixaCromodinâmicaForça nuclear forteGlúon10411/r71,4 × 10-15 mEletrodinâmicaForça eletromagnéticaFóton10391/r2infinitoFlavordinâmicaForça nuclear fracaBósons W e Z10291/r5 até 1/r710-18 mGeometrodinâmicaForça gravitacionalgráviton101/r2infinito$

Teoria	Interação	mediador	Magnitude relativa	Comportamento	Faixa
Cromodinâmica	Força nuclear forte	Glúon	1041	1/r7	1,4 × 10-15 m
Eletrodinâmica	Força eletromagnética	Fóton	1039	1/r2	infinito
Flavordinâmica	Força nuclear fraca	Bósons W e Z	1029	1/r5 até 1/r7	10-18 m
Geometrodinâmica	Força gravitacional	gráviton	10	1/r2	infinito

$G* = OPERADOR DE DIMENSÕES DE GRACELI.$

DIMENSÕES DE GRACELI SÃO TODA FORMA DE TENSORES, ESTRUTURAS, ENERGIAS, ACOPLAMENTOS, , INTERAÇÕES DE CAMPOS E ENERGIAS, DISTRIBUIÇÕES ELETRÔNICAS, ESTADOS FÍSICOS, ESTADOS QUÂNTICOS, ESTADOS FÍSICOS DE ENERGIAS DE GRACELI, E OUTROS.

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

MECÂNICA GRACELI GENERALIZADA - QUÂNTICA TENSORIAL DIMENSIONAL RELATIVISTA DE INTERAÇÕES DE CAMPOS. EM ;

MECÂNICA GRACELI REPRESENTADA POR TRANSFORMADA.

dd = dd [G] = DERIVADA DE DIMENSÕES DE GRACELI.

- [ $G*$ $\hbar$ $.$ $\psi$

G { f [dd]} ´[d] $\psi ^{*}$ $\hbar$ $\psi$ $f [d] G*$ $T_{\mu \nu }$ $\psi$ $dd [G]$

O ESTADO QUÂNTICO DE GRACELI

- [ $G*$ $\hbar$ $.$ $\psi$ $\left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigg )},$ ]

$G* = DIMENSÕES DE GRACELI TAMBÉM ESTÁ RELACIONADO COM INTERAÇÕES DE ENERGIAS, QUÂNTICAS, RELATIVÍSTICAS, , E INTERAÇÕES DE CAMPOS.$

o tensor energia-momento $T_{\mu \nu }$ é aquele de um campo eletromagnético,

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as $ψ$ or $Ψ$ (Greek psi), are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations (see classical field theory for background).

In the Schrödinger picture, the wave function or field is the solution to the Schrödinger equation; $i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi$ / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

one of the postulates of quantum mechanics. All relativistic wave equations can be constructed by specifying various forms of the Hamiltonian operator Ĥ describing the quantum system. Alternatively, Feynman's path integral formulation uses a Lagrangian rather than a Hamiltonian operator.

More generally – the modern formalism behind relativistic wave equations is Lorentz group theory, wherein the spin of the particle has a correspondence with the representations of the Lorentz group.^[1]

History

[edit]

Early 1920s: Classical and quantum mechanics

[edit]

The failure of classical mechanics applied to molecular, atomic, and nuclear systems and smaller induced the need for a new mechanics: quantum mechanics. The mathematical formulation was led by De Broglie, Bohr, Schrödinger, Pauli, and Heisenberg, and others, around the mid-1920s, and at that time was analogous to that of classical mechanics. The Schrödinger equation and the Heisenberg picture resemble the classical equations of motion in the limit of large quantum numbers and as the reduced Planck constant $ħ$ , the quantum of action, tends to zero. This is the correspondence principle. At this point, special relativity was not fully combined with quantum mechanics, so the Schrödinger and Heisenberg formulations, as originally proposed, could not be used in situations where the particles travel near the speed of light, or when the number of each type of particle changes (this happens in real particle interactions; the numerous forms of particle decays, annihilation, matter creation, pair production, and so on).

Late 1920s: Relativistic quantum mechanics of spin-0 and spin-¹/₂ particles

[edit]

A description of quantum mechanical systems which could account for relativistic effects was sought for by many theoretical physicists from the late 1920s to the mid-1940s.^[2] The first basis for relativistic quantum mechanics, i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called the Klein–Gordon equation:

-\hbar ^{2}{\frac {\partial ^{2}\psi }{\partial t^{2}}}+(\hbar c)^{2}\nabla ^{2}\psi =(mc^{2})^{2}\psi \,,

(1)

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

by inserting the energy operator and momentum operator into the relativistic energy–momentum relation:

E^{2}-(pc)^{2}=(mc^{2})^{2}\,,

(2)

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

The solutions to (1) are scalar fields. The KG equation is undesirable due to its prediction of negative energies and probabilities, as a result of the quadratic nature of (2) – inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the Schrödinger equation) was still of importance. Nevertheless, (1) is applicable to spin-0 bosons.^[3]

Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the fine structure in the Hydrogen spectral series. The mysterious underlying property was spin. The first two-dimensional spin matrices (better known as the Pauli matrices) were introduced by Pauli in the Pauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in magnetic fields, but this was phenomenological. Weyl found a relativistic equation in terms of the Pauli matrices; the Weyl equation, for massless spin-⁠1/2⁠ fermions. The problem was resolved by Dirac in the late 1920s, when he furthered the application of equation (2) to the electron – by various manipulations he factorized the equation into the form:

\left({\frac {E}{c}}-{\boldsymbol {\alpha }}\cdot \mathbf {p} -\beta mc\right)\left({\frac {E}{c}}+{\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta mc\right)\psi =0\,,

(3A)

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

and one of these factors is the Dirac equation (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices $α$ and $β$ in a relativistic wave equation, and explained the fine structure of hydrogen. The solutions to (3A) are multi-component spinor fields, and each component satisfies (1). A remarkable result of spinor solutions is that half of the components describe a particle while the other half describe an antiparticle; in this case the electron and positron. The Dirac equation is now known to apply for all massive spin-⁠1/2⁠ fermions. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.

Although a landmark in quantum theory, the Dirac equation is only true for spin-⁠1/2⁠ fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular – not all physicists were comfortable with the "Dirac sea" of negative energy states).

1930s–1960s: Relativistic quantum mechanics of higher-spin particles

[edit]

The natural problem became clear: to generalize the Dirac equation to particles with any spin; both fermions and bosons, and in the same equations their antiparticles (possible because of the spinor formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by van der Waerden in 1929), and ideally with positive energy solutions.^[2]

This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of (3A):

\left({\frac {E}{c}}+{\boldsymbol {\alpha }}\cdot \mathbf {p} -\beta mc\right)\psi =0\,,

(3B)

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $ψ$ is a spinor field now with infinitely many components, irreducible to a finite number of tensors or spinors, to remove the indeterminacy in sign. The matrices $α$ and $β$ are infinite-dimensional matrices, related to infinitesimal Lorentz transformations. He did not demand that each component of 3B satisfy equation (2); instead he regenerated the equation using a Lorentz-invariant action, via the principle of least action, and application of Lorentz group theory.^[4]^[5]

Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939) see Duffin–Kemmer–Petiau algebra. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana's, as spinors were new mathematical tools in the early twentieth century, although Majorana's paper of 1932 was difficult to fully understand; it took Pauli and Wigner some time to understand it, around 1940.^[2]

Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors $A$ and $B$ , symmetric in all indices, for a massive particle of spin $n + 1 ⁄ 2$ for integer $n$ (see Van der Waerden notation for the meaning of the dotted indices):

p_{\gamma {\dot {\alpha }}}A_{\epsilon _{1}\epsilon _{2}\cdots \epsilon _{n}}^{{\dot {\alpha }}{\dot {\beta }}_{1}{\dot {\beta }}_{2}\cdots {\dot {\beta }}_{n}}=mcB_{\gamma \epsilon _{1}\epsilon _{2}\cdots \epsilon _{n}}^{{\dot {\beta }}_{1}{\dot {\beta }}_{2}\cdots {\dot {\beta }}_{n}}

(4A)

/ $\psi ^{*}$ [ $\psi ^{*}$

T_{\mu \nu }

$-i\hbar {\frac {\partial }{\partial x_{n}}}$

p^{\gamma {\dot {\alpha }}}B_{\gamma \epsilon _{1}\epsilon _{2}\cdots \epsilon _{n}}^{{\dot {\beta }}_{1}{\dot {\beta }}_{2}\cdots {\dot {\beta }}_{n}}=mcA_{\epsilon _{1}\epsilon _{2}\cdots \epsilon _{n}}^{{\dot {\alpha }}{\dot {\beta }}_{1}{\dot {\beta }}_{2}\cdots {\dot {\beta }}_{n}}

(4B)

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

where $p$ is the momentum as a covariant spinor operator. For $n = 0$ , the equations reduce to the coupled Dirac equations and $A$ and $B$ together transform as the original Dirac spinor. Eliminating either $A$ or $B$ shows that $A$ and $B$ each fulfill (1).^[2] The direct derivation of the Dirac-Pauli-Fierz equations using the Bargmann-Wigner operators is given in.^[6]

In 1941, Rarita and Schwinger focussed on spin-3⁄2 particles and derived the Rarita–Schwinger equation, including a Lagrangian to generate it, and later generalized the equations analogous to spin $n + 1 ⁄ 2$ for integer $n$ . In 1945, Pauli suggested Majorana's 1932 paper to Bhabha, who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in (3A) and (3B) by an arbitrary constant, subject to a set of conditions which the wave functions must obey.^[7]

Finally, in the year 1948 (the same year as Feynman's path integral formulation was cast), Bargmann and Wigner formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): the Bargmann–Wigner equations.^[2]^[8] In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by H. Joos and Steven Weinberg, the Joos–Weinberg equation. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.^[1]^[9]^[10]

1960s–present

[edit]

The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of the present-day research because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present.^[5]

Linear equations

[edit]

The following equations have solutions which satisfy the superposition principle, that is, the wave functions are additive.

Throughout, the standard conventions of tensor index notation and Feynman slash notation are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the indexed quantities. The wave functions are denoted $ψ$ , and $\partial μ$ are the components of the four-gradient operator.

In matrix equations, the Pauli matrices are denoted by $σ μ$ in which $μ = 0, 1, 2, 3$ , where $σ 0$ is the $2 \times 2$ identity matrix: $\sigma ^{0}={\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$ and the other matrices have their usual representations. The expression $\sigma ^{\mu }\partial _{\mu }\equiv \sigma ^{0}\partial _{0}+\sigma ^{1}\partial _{1}+\sigma ^{2}\partial _{2}+\sigma ^{3}\partial _{3}$ / $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

is a $2 \times 2$ matrix operator which acts on 2-component spinor fields.

In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.

Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry.

Overview

[edit]

In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors. One method for dealing with this problem is to require that M have a spin structure.^[1]^[2]^[3] This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M vanishes. Furthermore, if w₂(M) = 0, then the set of the isomorphism classes of spin structures on M is acted upon freely and transitively by H¹(M, Z₂) . As the manifold M is assumed to be oriented, the first Stiefel–Whitney class w₁(M) ∈ H¹(M, Z₂) of M vanishes too. (The Stiefel–Whitney classes w_i(M) ∈ Hⁱ(M, Z₂) of a manifold M are defined to be the Stiefel–Whitney classes of its tangent bundle TM.)

The bundle of spinors π_S: S → M over M is then the complex vector bundle associated with the corresponding principal bundle π_P: P → M of spin frames over M and the spin representation of its structure group Spin(n) on the space of spinors Δ_n. The bundle S is called the spinor bundle for a given spin structure on M.

A precise definition of spin structure on manifold was possible only after the notion of fiber bundle had been introduced; André Haefliger (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and Max Karoubi (1968) extended this result to the non-orientable pseudo-Riemannian case.^[4]^[5]

Spin structures on Riemannian manifolds

[edit]

Definition

[edit]

A spin structure on an orientable Riemannian manifold $(M,g)$ with an oriented vector bundle $�$ is an equivariant lift of the orthonormal frame bundle $P_{\operatorname {SO} }(E)\rightarrow M$ with respect to the double covering $\rho :\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)$ . In other words, a pair $(P_{\operatorname {Spin} },\phi )$ is a spin structure on the SO(n)-principal bundle $\pi :P_{\operatorname {SO} }(E)\rightarrow M$ when

\pi _{P}:P_{\operatorname {Spin} }\rightarrow M

is a principal Spin(n)-bundle over

�

, and

\phi :P_{\operatorname {Spin} }\rightarrow P_{\operatorname {SO} }(E)

is an equivariant 2-fold covering map such that

$\pi \circ \phi =\pi _{P}\quad$ and $\quad \phi (pq)=\phi (p)\rho (q)\quad$ for all $p\in P_{\operatorname {Spin} }$ and $q\in \operatorname {Spin} (n)$ .

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

Two spin structures $(P_{1},\phi _{1})$ and $(P_{2},\phi _{2})$ on the same oriented Riemannian manifold $f:P_{1}\rightarrow P_{2}$ are called "equivalent" if there exists a Spin(n)-equivariant map $f:P_{1}\rightarrow P_{2}$ such that

\phi _{2}\circ f=\phi _{1}\quad

and

\quad f(pq)=f(p)q\quad

for all

p\in P_{1}

and

q\in \operatorname {Spin} (n)

In this case $\phi _{1}$ and $\phi _{2}$ are two equivalent double coverings.

The definition of spin structure on $(M,g)$ as a spin structure on the principal bundle $P_{\operatorname {SO} }(E)\rightarrow M$

$is due to André Haefliger (1956).$

Obstruction

[edit]

Haefliger^[1] found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold (M,g). The obstruction to having a spin structure is a certain element [k] of H²(M, Z₂) . For a spin structure the class [k] is the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M. Hence, a spin structure exists if and only if the second Stiefel–Whitney class w₂(M) ∈ H²(M, Z₂) of M vanishes.

Spin structures on vector bundles

[edit]

Let M be a paracompact topological manifold and E an oriented vector bundle on M of dimension n equipped with a fibre metric. This means that at each point of M, the fibre of E is an inner product space. A spinor bundle of E is a prescription for consistently associating a spin representation to every point of M. There are topological obstructions to being able to do it, and consequently, a given bundle E may not admit any spinor bundle. In case it does, one says that the bundle E is spin.

This may be made rigorous through the language of principal bundles. The collection of oriented orthonormal frames of a vector bundle form a frame bundle P_SO(E), which is a principal bundle under the action of the special orthogonal group SO(n). A spin structure for P_SO(E) is a lift of P_SO(E) to a principal bundle P_Spin(E) under the action of the spin group Spin(n), by which we mean that there exists a bundle map $\phi$ $\phi$ : P_Spin(E) → P_SO(E) such that

\phi (pg)=\phi (p)\rho (g)

, for all p ∈ P_Spin(E) and g ∈ Spin(n),

where ρ : Spin(n) → SO(n) is the mapping of groups presenting the spin group as a double-cover of SO(n).

In the special case in which E is the tangent bundle TM over the base manifold M, if a spin structure exists then one says that M is a spin manifold. Equivalently M is spin if the SO(n) principal bundle of orthonormal bases of the tangent fibers of M is a Z₂ quotient of a principal spin bundle.

If the manifold has a cell decomposition or a triangulation, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the tangent bundle over the 1-skeleton that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a Whitney sum with a trivial line bundle.

Obstruction and classification

[edit]

For an orientable vector bundle $\pi _{E}:E\to M$ a spin structure exists on $�$ if and only if the second Stiefel–Whitney class $w_{2}(E)$ vanishes. This is a result of Armand Borel and Friedrich Hirzebruch.^[6] Furthermore, in the case $E\to M$ is spin, the number of spin structures are in bijection with $H^{1}(M,\mathbb {Z} /2)$ . These results can be easily proven^[7]^{pg 110-111} using a spectral sequence argument for the associated principal $\operatorname {SO} (n)$ -bundle $P_{E}\to M$ . Notice this gives a fibration

$\operatorname {SO} (n)\to P_{E}\to M$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

hence the Serre spectral sequence can be applied. From general theory of spectral sequences, there is an exact sequence

$0\to E_{3}^{0,1}\to E_{2}^{0,1}\xrightarrow {d_{2}} E_{2}^{2,0}\to E_{3}^{2,0}\to 0$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

where

${\begin{aligned}E_{2}^{0,1}&=H^{0}(M,H^{1}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\\E_{2}^{2,0}&=H^{2}(M,H^{0}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{2}(M,\mathbb {Z} /2)\end{aligned}}$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

In addition, $E_{\infty }^{0,1}=E_{3}^{0,1}$ and $E_{\infty }^{0,1}=H^{1}(P_{E},\mathbb {Z} /2)/F^{1}(H^{1}(P_{E},\mathbb {Z} /2))$ for some filtration on $H^{1}(P_{E},\mathbb {Z} /2)$ , hence we get a map

$H^{1}(P_{E},\mathbb {Z} /2)\to E_{3}^{0,1}$

giving an exact sequence

$H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\to H^{2}(M,\mathbb {Z} /2)$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

Now, a spin structure is exactly a double covering of $P_{E}$ fitting into a commutative diagram

${\begin{matrix}\operatorname {Spin} (n)&\to &{\tilde {P}}_{E}&\to &M\\\downarrow &&\downarrow &&\downarrow \\\operatorname {SO} (n)&\to &P_{E}&\to &M\end{matrix}}$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

where the two left vertical maps are the double covering maps. Now, double coverings of $P_{E}$ are in bijection with index $2$ subgroups of $\pi _{1}(P_{E})$ , which is in bijection with the set of group morphisms ${\text{Hom}}(\pi _{1}(E),\mathbb {Z} /2)$ . But, from Hurewicz theorem and change of coefficients, this is exactly the cohomology group $H^{1}(P_{E},\mathbb {Z} /2)$ . Applying the same argument to $\operatorname {SO} (n)$ , the non-trivial covering $\operatorname {Spin} (n)\to \operatorname {SO} (n)$ corresponds to $1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)=\mathbb {Z} /2$ , and the map to $H^{2}(M,\mathbb {Z} /2)$ is precisely the $w_{2}$ of the second Stiefel–Whitney class, hence $w_{2}(1)=w_{2}(E)$ . If it vanishes, then the inverse image of $1$ under the map

$H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

is the set of double coverings giving spin structures. Now, this subset of $H^{1}(P_{E},\mathbb {Z} /2)$ can be identified with $H^{1}(M,\mathbb {Z} /2)$ , showing this latter cohomology group classifies the various spin structures on the vector bundle $E\to M$ . This can be done by looking at the long exact sequence of homotopy groups of the fibration

$\pi _{1}(\operatorname {SO} (n))\to \pi _{1}(P_{E})\to \pi _{1}(M)\to 1$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

and applying ${\text{Hom}}(-,\mathbb {Z} /2)$ , giving the sequence of cohomology groups

$0\to H^{1}(M,\mathbb {Z} /2)\to H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)$

/ $\psi ^{*}$ [ $\psi ^{*}$ $T_{\mu \nu }$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$

Because $H^{1}(M,\mathbb {Z} /2)$ is the kernel, and the inverse image of $1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)$ is in bijection with the kernel, we have the desired result.

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MECÂNICA GRACELI GENENRALIZADA - quântica TENSORIAL DIMENSIONAL RELATTIVISTA DE CAMPOS - [1.2015]

History

Early 1920s: Classical and quantum mechanics

Late 1920s: Relativistic quantum mechanics of spin-0 and spin-¹/₂ particles