Non-abelian gauge symmetry for fields in phase space: a realization of the Seiberg-Witten non-abelian gauge theory

TL;DR

This paper extends the Seiberg-Witten non-abelian gauge theory to phase space, formulating an SU(2) gauge field for the Dirac equation and exploring its application to nucleons.

Contribution

It introduces a non-abelian gauge formalism in phase space for the Dirac field, generalizing the Seiberg-Witten approach to SU(2) symmetry.

Findings

Derived a non-abelian field strength in phase space.

Formulated an SU(2) gauge theory for the Dirac field in phase space.

Discussed potential applications to nucleon physics.

Abstract

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

Equations

$$\left[{\hat{x}}^{\mu },{\hat{x}}^{\nu }\right]=i{\theta }^{\mu \nu },$$

$$\hat{f}\equiv \hat{W}[f]=\int \frac{d^{d}k}{(2\pi {)}^d}\tilde{f}(k)e_k(\hat{x}),$$

$$f(x)=\int \frac{d^{d}k}{(2\pi {)}^d}f(k)e_k(x).$$

$$f(x)\equiv {\hat{W}}^{-1}[\hat{f}]=\int \frac{d^{d}k}{(2\pi {)}^d}{e}^{-i{k}_{\mu }{x}^{\mu }}\mathrm{Tr}(\hat{f}{e}_k(\hat{x})),$$

$$f\star g={\hat{W}}^{-1}[\hat{W}[f]\hat{W}[g]],f,g\in {\mathcal{A}}_{\theta }({\mathbb{R}}^d),$$

$$(f\star g)(x)=f(x)e^{\frac{i}{2}{\overleftarrow{\partial }}_{\mu }{\theta }^{\mu \nu }{\overrightarrow{\partial }}_{\nu }}g(x).$$

$$F_{\theta }^{\mu \nu }={\partial }_{\nu }{A}^{\mu }-{\partial }_{\mu }{A}^{\nu }-i\{A^{\mu },A^{\nu }{\}}_M,$$

$${\psi }_i^{\prime }(x)=U_{ij}(x;\lambda )\star {\psi }_j(x)\text{and}{\overline{\psi }}_i^{\prime }(x)={\overline{\psi }}_j(x)\star U_{ji}^{†}(x;\lambda ),$$

$$F_{\theta }^{\mu \nu }={\partial }_{\nu }{A}^{\mu }-{\partial }_{\mu }{A}^{\nu }-i[A^{\mu },A^{\nu }{]}_M,$$

$$\mathcal{L}=\frac{1}{4}{F}_{\theta }^{\mu \nu }{F}_{\theta \mu \nu }.$$

$$\eta=\left(\begin{array}[]{cc}0&I\\ -I&0\end{array}\right),$$

$$\Lambda ={\eta }_{ab}\frac{\overleftarrow{\partial }}{\partial {\omega }^a}\frac{\overrightarrow{\partial }}{\partial {\omega }^b},$$

$$\Omega (f\Lambda ,g\Lambda )$$

$$F=f(\omega )\star =f(\omega )e^{\frac{i}{2}{\overleftarrow{\partial }}_a\hslash {\eta }^{ab}{\overrightarrow{\partial }}_b},$$

$$P^{\mu }$$

$$Q^{\mu }$$

$$\frac{-{\hslash }^2}{4}\frac{{\partial }^2\varphi (p,q)}{\partial q^{\mu }\partial q_{\mu }}-i\hslash p^{\mu }\frac{\partial \varphi (p,q)}{\partial q^{\mu }}+(p^{\mu }{p}_{\mu }-m^2)\varphi (p,q)=0.$$

$$f_W(q,p)=\varphi (q,p)\star {\varphi }^{†}(q,p).$$

$$\psi (q,p)\to {\psi }^{\prime }(q,p)=S^{-1}\star \psi (q,p,),$$

$$S=\exp (-i\mathbf{\tau }\cdot \mathbf{\alpha }),$$

$$\psi (p,q)$$

$$S\star {\psi }^{\prime }$$

$$\frac{\partial Y^{-1}}{\partial x}=-Y^{-1}\frac{\partial Y}{\partial x}{Y}^{-1}.$$

$$S\star S^{-1}$$

$$G_{\mu }\star =p_{\mu }\star +iϵ(\mathbf{\tau }\cdot {\mathbf{B}}_{\mu })\star ,$$

$$G_{\mu }\star \psi (q,p)=S\star G_{\mu }^{{}^{\prime }}\star {\psi }^{\prime }(p,q).$$

$$G_{\mu }\star \psi (q,p)$$

$$T_9$$

$$\frac{\partial Y^{-1}}{\partial x}\star =-Y^{-1}\star \frac{\partial Y}{\partial x}\star Y^{-1}\star .$$

$$T_9$$

Full text

Jan. 28, 2018

I Introduction

The development of quantum mechanics from classical mechanics initiated with the experimental observation that position and momentum coordinates do not commute. This had a tremendous impact on the development of dynamics of a broad variety of areas, including quantum corrections to statistical mechanics in phase space, a program that had its origin with Wigner Wigner . In addition, there was the question about non-commutation of position coordinates, a conjecture raised by Heisenberg Heisenberg , and first developed by Snyder Snyder , studying representations of Lie-groups. This was followed by advances in non-commutative geometry, having as a realization the algebraic structure of the Wigner approach, based on the non-commutative Moyal (star)-product in phase space Moyal ; Hillery ; zabo1 ; Grosse .

Then the problem of a non-commutative quantum field came about. In this case, the abelian gauge symmetry for non-commutative fields was addressed consistently first by Seiberg and Witten SeiberWitten , that used this for string theory. The usual gauge-field strength is then generalized to a spin-one non-commutative field theory and the quantum field approach gives rise to a mixture of infrared and ultraviolet divergences that usually breakdown renormalizability zabo1 ; Minwalla .

The difficulties with the gauge non-commutative fields have been handled Langmann ; Magnen , and numerous applications of non-commutative theories and its algebraic structure have been achieved, considering, for instance: the field theory of the standard model of particle physics, including the development of the non-commutative non-abelian theory zabo1 ; Mariz ; Costa ; Gurau ; the gravitational theory Kalau ; Kastler ; supersymmetry Girotti ; quantum Hall effect Belissard ; unification of gravity and quantum field theory Yu ; and the pair-creation of neutral Dirac particles in $(1+2)d$ non-commutative space-time have been investigated Hammil .

A realization of the Seiberg-Witten-gauge theory has been derived by exploring a U(1)-unitary representation of the Poincaré space-time symmetry in phase space Amorim0 , giving rise to a symplectic quantum electrodynamics where the propagators are associated with the Wigner function Amorim19 . The non-commutativity, in this case, is among the two sectors of the symplectic manifold; that is, among coordinates and momenta. In such a symplectic realization, the field strength is given by the algebraic non-commutative structure of the Moyal (star)-product Moyal ; Hillery . Other representations, such as spin zero and spin 1/2, have been explored for the Galilei and Poincaré groups Oliveira ; Ronni , leading to the Schrödinger equation in phase space, in the non-relativistic realm, as well as for the Klein-Gordon and Dirac equation in a relativistic phase space. The Wigner function is given by $f_{w}(q,p)=~{}\psi(q,p)\star\psi^{{\dagger}}(q,p)$, where $\psi(q,p)$ stands for the wave function in phase space called the quasi-amplitudes of probability and “$\star$” is the Moyal product in this phase space.

One basic motivation for such developments is the calculation of the Wigner function for the states of relativistic particles described by a quantum field Amorim0 ; Amorim19 . This is important for analysing the statistical nature (such as chaoticity) of quantum states, with plain measurement in some systems wig3 ; wig4 ; 2gal1 ; 2davido1 ; 2kha1 . Despite the importance, the direct application of the Wigner formalism is not a straightforward procedure. For instance, considering interaction, neither the Liouville-von Neumann equation is trivially generalized for a quantum field, nor the gauge symmetry is a simple task to be taken into consideration, since the Wigner function is a Real function. These problems have led to looking for solution in a representation formalism based on unitary transformations in phase space, such that the state, the quasi-amplitude of probability, is consistently associated with the Wigner function. This has been accomplished Oliveira ; Ronni , and in this approach, the Seiberg-Witten-gauge theory arises naturally by studying gauge transformations of the type $\psi(q,p)\rightarrow e^{-i\lambda(q,p)}\star\psi(q,p)$, leading to a non-commutative-like electrodynamics, such that the propagators, describing bosons and fermions, are related to the Wigner function Amorim0 . Applications of such results include: the interaction between the Dirac equation with an external electromagnetic field in phase space Ronni001 ; analysis of the Wigner function for the Landau problem Ronni002 ; evaluation of the negativity of the Wigner function for a system defined by the sum of Hénon-Heiles potential and Hydrogen atom Cruz .

Despite these advances with the field theory in phase phase, the non-abelian gauge symmetry has not been developed for fields in phase space. This problem is addressed here, first by following a heuristic presentation of the Seiberg-Witten approach, to include a non-abelian gauge group in the non-commutative field strength in phase space. This analysis is carried out in Section 2, where the notation is fixed and some aspects of a non-commutative field are outlined. In Section 3, starting with the Lagrangian density for the free Dirac field in phase space, and considering the SU(2) as the non-abelian gauge group, a field strength is derived, corresponding to a realization of the Lagrangian and the field strength discussed in Section 2. In Section 4, an application to the nucleon in an external field is presented. Final concluding remarks are presented in Section 5. In Appendix A, some aspects of the symplectic field theory are outlined in order to fix the notation. In Appendix B, some details of the calculation for the abelian symplectic gauge field is presented for the sake of completeness and for showing, as a guide, some steps and properties used in Section 3.

II Gauge theory for non-commutative fields

The non-commutative plane or the Groenewold-Moyal (GM) plane is defined by the following procedure. Consider the manifold $\mathbb{R}^{d}$ with commutative coordinates $x_{\mu}$, $\mu=1,...,d$, i.e. $\left[x^{\mu},x^{\nu}\right]=0.$ The space of complex functions is then introduced: $\mathcal{A}_{0}(\mathbb{R}^{d})=\{f:\mathbb{R}^{d}\rightarrow\mathbb{C}\}$, such that $\mathcal{A}_{0}(\mathbb{R}^{d})$ is an associative algebra with multiplication $f\cdot g(x):=f(x)g(x)$. There is then a 1:1 map between the manifold $\mathbb{R}^{d}$ and the algebra $\mathcal{A}_{0}(\mathbb{R}^{d})$.

A deformed algebra of operators $\mathbb{R}_{\theta}^{d}$ is generated by the $d$ operators $\hat{x}^{\mu}$ with the deformed relations

[TABLE]

with \theta^{\mu\nu}=-\theta^{\nu\mu}\ being real constants. The Weyl quantization is a mapping from elements of the complex vector space $\mathcal{A}_{0}(\mathbb{R}^{d})$ to elements of the algebra $\mathbb{R}_{\theta}^{d}$ through the Weyl map $\hat{W}$, which is defined by $\hat{W}:\mathcal{A}_{0}(\mathbb{R}^{d})\rightarrow\mathbb{R}_{\theta}^{d}$, such that $e_{k}(x)=\exp(ik_{\mu}x^{\mu})\mapsto e_{k}(\hat{x})\equiv\exp(ik_{\mu}\hat{x}^{\mu})$ and with

[TABLE]

where $~{}f(k)$ is the Fourier transform of $~{}f(x)$; i.e.

[TABLE]

The operator $\hat{W}[f]$ is a non-commutative field.

The inverse of the Weyl map, $W,$ is called the Wigner function, which is defined by $\hat{W}^{-1}:\mathbb{R}_{\theta}^{d}\rightarrow\mathcal{A}_{0}(\mathbb{R}^{d})$; or $\hat{f}\mapsto\hat{W}^{-1}[\hat{f}]$, such that

[TABLE]

where $\mathrm{Tr}$ is the trace. The map $W$ provides a vector space (not an algebra) isomorphism. Then an algebra $\mathcal{A}_{\theta}(\mathbb{R}^{d})$ is introduced with the Moyal product or $\star$-product, given by

[TABLE]

such that for $\theta^{\mu\nu}\mapsto 0,$ the product $(f\star g)(x)=f(x)\star g(x)\mapsto f(x)g(x)$. This Moyal product is associative, but it is not commutative. Then $\mathcal{A}_{\theta}(\mathbb{R}^{d})$ is a non-commutative algebra defined from the commutative plane $\mathcal{A}_{0}(\mathbb{R}^{d}).$ Then the algebra $\mathcal{A}_{\theta}(\mathbb{R}^{d})$ is called a non-commutative plane or the Groenewold-Moyal (GM) plane. As an example, for $\ C^{\infty}$ functions in $\mathcal{A}_{\theta}(\mathbb{R}^{d})$, a realization of the Moyal product is given by

[TABLE]

This structure is the starting point for developing the non-commutative field theory; and considering the association of non-commutative geometry with string theory, Seiberg and Witten SeiberWitten studied a string dynamics described by a minimally coupled supersymmetric gauge field in a non-commutative space. The result is a generalized gauge field $A^{\mu}$ with an antisymmetric field strength given by

[TABLE]

where $\{A^{\mu},A^{\nu}\}_{M}$ is the Moyal bracket, given by $\{f,g\}_{M}=f\star g-g\star f$. It is to be noted that \theta^{\mu\nu}\mapsto 0,\ \ \{f,g\}_{M}\mapsto 0,\ i.e., $F_{\theta}^{\mu\nu}\mapsto F^{\mu\nu}=\partial_{\nu}A^{\mu}-\partial_{\mu}A^{\nu}$; that is, the usual electrodynamics (the U$(1)$ gauge-group theory) is recovered. In this case, the starting point is the (3+1)-Minkowski space, $\mathbb{M}$, with $\mu,\nu=0,1,2,3$, and the metric $g$ being such that $diag(g)=(1,-1,-1,-1)$. This analysis provides the hint for writing the Seiberg-Witten theory considering non-abelian fields, with the non-abelian group characterized by the structure constants $c_{rsl}$ zabo1 .

The phase transformation of a non-commutative field can be written as $\psi^{\prime}(x)=U(x;\lambda)\star\psi(x),$ where $U(x;\lambda)=e^{i\lambda(x)}$, with $\lambda(x)$ being a real-valued operator of the space-time coordinates. The transformation $U(x;\lambda)$ is unitary. Considering components of $\psi_{j}(x)$, $j=1,2,...,N$ , the transformation $U$ is an $N\times N$ matrix in the indices $i$ and $j$, such that

[TABLE]

where repeated Latin indices are summed. As $U_{ij}(x;\lambda)$ is connected to identity, we write $U(x;\lambda)=e^{-i\xi\lambda(x)}=e^{-i\xi\Lambda_{r}(x)t_{r}},$ where $\lambda_{r}(x)$ are real-valued functions of space-time coordinates and $\xi$ is a constant fixing the units. The quantities $t_{j}$ are operators of the gauge group, satisfying the Lie algebra $[t_{r},t_{s}]=c_{rsl}t_{l}$, where $c_{rsl}$ are the structure constants of the group, with $r,s,l=1,2,...,N$, and $N$ stands for the number of group generators $t_{r}$, taken in the adjoint representation. Then each matrix $t_{r}$ is given in the form $(t_{r})_{sl}=(c_{rsl})$.

The non-abelian Seiberg-Witten gauge theory is carried out by using specific non-commutative field theories. It is expected that for $c_{rsl}=0$, the field strength reduces to eq. (7). The natural candidate is a field strength given by

[TABLE]

where $[A^{\mu},A^{\nu}]_{M}=A^{\mu}\star A^{\nu}-A^{\nu}\star A^{\mu}$ (it is important to emphasize that the fields $A^{\mu}$ are matrices, as usual). Under a gauge transformation, the gauge field transforms as $A_{\mu}^{{}^{\prime}}=A_{\mu}+[iA_{\mu},\lambda]_{M}-\frac{i}{\xi}\partial_{\mu}\lambda$. In addition, observe that for $c_{rsl}=0,$  $[A^{\mu},A^{\nu}]_{M}\mapsto\{f,g\}_{M}.$ Consistently, for non-zero structure constants, and for $\theta^{\mu\nu}\mapsto 0,\ \ \ [A^{\mu},A^{\nu}]_{M}\mapsto[A^{\mu},A^{\nu}]=A^{\mu}A^{\nu}-A^{\nu}A^{\mu}.$ With this field strength, the simplest non-abelian Lagrangian density reads

[TABLE]

It has to be noted that there is no Moyal product in the product of two field strength, as usual in non-commutative theories zabo1 .

In the next section, these results are demonstrated explicitly for the symplectic field theory, a non-commutative-like field theory in phase space, associated with the Wigner function formalism. In this approach, the starting manifold is the cotangent-bundle of the Minkowski space, $T^{\ast}\mathbb{M}$, where each point is specified by the coordinates $(q^{\mu},p^{\mu})\in T^{\ast}\mathbb{M}=\mathbb{M}\times\mathbb{M}$. The 8-dimensional space $T^{\ast}\mathbb{M}$ is equipped with a 2-form $\Omega=dq^{\mu}\wedge dp_{\mu}$, called the symplectic form. The space $T^{\ast}\mathbb{M}$ endowed with this symplectic structure is called the symplectic (or phase) space, and will be denoted by $\Gamma$. Let us define a vector in this manifold by $\omega=({\omega}^{1},{\omega}^{2},\ldots,{\omega}^{8})$, with $\omega^{1}=q^{0},\omega^{2}=q^{1},\omega^{3}=q^{2},\omega^{4}=q^{3},$ $\omega^{5}=p^{0},\omega^{6}=p^{1},\omega^{7}=p^{2},\omega^{8}=p^{3}$ with $q=(q^{0},\mathbf{q})$ and $p=(p^{0},\mathbf{p})$ being vectors in $\mathbb{M}$. The symplectic metric matrix in $\Gamma$ is given by (${\eta}_{ab})$, $a,b=1,...,8$,

[TABLE]

where $I$ is a $4\otimes 4$ unit matrix. We define the following vector fields, operators on $C^{\infty}(T^{\ast}M)$,

[TABLE]

such that for $C^{\infty}$ functions, $f(\omega)=f(q,p)$ and $g(\omega)=g(q,p),$ we have

[TABLE]

where $\{f,g\}=\frac{\partial f}{\partial q^{\mu}}\frac{\partial g}{\partial p_{\mu}}-\frac{\partial f}{\partial p^{\mu}}\frac{\partial g}{\partial q_{\mu}}$ is the Poisson bracket.

In this structure, a Hilbert space is introduced and used as a carrier space for the Poincaré symmetry. The basic unitary operators are given by

[TABLE]

where $\hbar$ is the Planck constant and the star-notation for the Moyal product is used as $\star=\exp^{\frac{i}{2}\overleftarrow{\partial}_{a}\hbar\eta^{ab}\overrightarrow{\partial}_{b}}$, with $\theta^{ab}=\hbar\eta^{ab}$.

Representations for the Poincaré symmetry are taken by considering the function $f$ to be components of $q$ and $p$. Explicitly, we have

[TABLE]

These operators satisfy the Heinsenberg condition, i.e. $[Q^{\mu},P^{\nu}]=i\hbar g^{\mu\nu}$ . Therefore, Lorentz transformations are introduced by defining the generators $M_{\mu\nu}=Q^{\mu}P^{\nu}-P^{\nu}Q^{\mu}$. It is simple to verify the physical consistency of the representation, $P$ is taken as the generator of translation in the Minkowski space. Then we have for $Q$ and $P$ the transformation rules of position and momentum, respectively. Indeed: $U(a)=\exp[-ia_{\mu}{P}^{\mu}]$ such that $U(a){Q}_{\mu}U(a)^{\dagger}={Q}_{\mu}+a_{\mu}$ and $U(a)\psi(q,p)=e^{ia_{\mu}p^{\mu}}\psi(q^{\mu}+a^{\mu},p)$, with $[\hat{P}_{\mu},\hat{P}_{\nu}]=0$.

An equation of motion describing spin-zero particles is obtained by using the Casimir invariant $P^{2}=m^{2}$, leading to $P^{\mu}P_{\mu}\phi(p,q)=m^{2}\phi(p,q)$, that is given explicitly by

[TABLE]

This is a Klein-Gordon-like equation written in phase space. The field $\phi(q,p)$ is associated with a Wigner function by Oliveira ; Amorim0

[TABLE]

This result is important to provide a full representation.

Some details for the Hilbert space in the phase space is presented in Appendix A and a representation for spin 1/2 particles is detailed in the Appendix B. Here it is important to mention that in this representation, the free field has already the content of a non-commutative theory due to the presence of the Moyal product defining the Lie-algebra representation. This aspect is central for providing the correct interpretation of the theory, when the connection of $\phi(q,p)$, the quasi-amplitude of probability, with the Wigner function is established, as given in eq. (14). An analysis of the gauge invariance following similar steps of the standard gauge theory are expected.

III Isospin in phase space

A local isotopic gauge transformation in phase space is such that

[TABLE]

where $S$ represents a unitary matrix $2\times 2$ with $SS^{-1}=S^{-1}S=1$, and $\psi(q,p)$ is a two-component wavefunction. In this case, $\psi(q,p)$ describes a field with isotopic spin $1/2$. The matrix $S$ is given as

[TABLE]

where $\bm{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})$, and $\bm{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ represent the Pauli matrices.

From the gauge transformation, eq. (15), one has

[TABLE]

Therefore, by multiplying the left of eq. (15) by $S\star$ leads to

[TABLE]

Expanding the star product into power series of $\hbar$, keeping terms of first order, we obtain

[TABLE]

where the scalar identity is used

[TABLE]

This leads to

[TABLE]

In analogy with the abelian case where the covariant derivative is $D_{\mu}{\star}=\left(p_{\mu}\star+ieA_{\mu}\star\right)$, for isospin the definition is

[TABLE]

where $\mathbf{B}_{\mu}=(B_{\mu}^{1},B_{\mu}^{2},B_{\mu}^{3})$. The invariance of local gauge transformation requires

[TABLE]

Using eq. (15) in eq. (21), the following transformation for $B_{\mu}$ is

[TABLE]

The first term on the right side of eq. (22) is considered separately leading to

[TABLE]

Here the following relation is used

[TABLE]

Using eq. (72), the first term of the eq. (23) becomes $S\star(p_{\mu}S^{-1})=p_{\mu}(S\star S^{-1})+\frac{i\hbar}{2}(\frac{\partial S}{\partial q^{\mu}})\star S^{-1}$, then

[TABLE]

Then eq. (22) becomes

[TABLE]

leading to

[TABLE]

In order to isolate the term $(\bm{\tau}\cdot\mathbf{B}^{\prime}_{\mu})$, multiply the above equation by $S$, which gives

[TABLE]

where $(\bm{\tau}\cdot\mathbf{B}^{\prime}_{\mu})\star S^{-1}\star S=(\bm{\tau}\cdot\mathbf{B}^{\prime}_{\mu})$ e $\frac{\partial S}{\partial q^{\mu}}\star S^{-1}\star S=\frac{\partial S}{\partial q^{\mu}}$. Multiplying eq. (25) by $S^{-1}\star$ from the left, leads to

[TABLE]

Therefore, the field $B_{\mu}$ obeys the gauge transformation given by eq. (15). In analogy with the electromagnetic field, where the relation $[D_{\mu}{\star},D_{\nu}{\star}]$ is used to find $\mathcal{F}_{\mu\nu}$, for the case of Isospin, this leads to

[TABLE]

where eq. (20) has been used. Now each term is calculated separately.

The first term becomes $\left[p_{\mu}\star,p_{\nu}\star\right]\psi(q,p)=0$. The second term is

[TABLE]

Using eq. (72) in Appendix B, the following relation is derived,

[TABLE]

In analogy with eq. (28), the third term of eq. (27) takes the form

[TABLE]

The last term is given by

[TABLE]

Hence, eq. (27) becomes

[TABLE]

As a consequence, a tensor strength for the isospin is defined as

[TABLE]

where $b_{\mu}=\bm{\tau}\cdot\mathbf{B}_{\mu}$. For the Isospin, using eq. (26) the tensor is transformed to the form

[TABLE]

Considering

[TABLE]

then $\mathcal{K}_{\mu\nu}\star$ has the form

[TABLE]

Multiplying by $S$ on the right-hand side leads to

[TABLE]

By multiplying by $S^{-1}\star$ on the left-hand side leads to

[TABLE]

Then eq. (32) becomes

[TABLE]

The Pauli matrices have the property

[TABLE]

A similar property is connected to the star product

[TABLE]

where $\mathbf{A}\cdot\bm{\sigma}$ is given by

[TABLE]

and the term $(\mathbf{B}\cdot\bm{\sigma})$, has the form

[TABLE]

Expanding the star product of eq. (36) leads to

[TABLE]

The first term is written as

[TABLE]

And the second and third terms in eq. (37) take the form

[TABLE]

Therefore, using these results, eq. (37) becomes

[TABLE]

where $\left(\mathbf{A}\times\mathbf{B}\right)_{\star\hat{k}}=\left(A_{i}\star B_{j}-A_{j}\star B_{i}\right)_{\hat{k}},$ with $\hat{i},\hat{j},\hat{k}=\hat{x},\hat{y},\hat{z}.$

The terms of eq. (34) become

[TABLE]

and

[TABLE]

Then eq. (34) is written as

[TABLE]

where

[TABLE]

The infinitesimal gauge transformation for eq. (16) is given by

[TABLE]

Then eq. (26) becomes

[TABLE]

Assuming a field with isotopic spin $1/2$, the final Lagrangian is

[TABLE]

describing an isospin in phase space.

As in the previous sections, the star product is expanded in power series of $\hbar$ up to first order. Then expanding eq. (15), the isotopic gauge transformation is given by

[TABLE]

The covariant derivative is defined as

[TABLE]

Requiring that the covariant derivative is over the local gauge leads to

[TABLE]

Therefore, the isotope field, $\mathbf{B^{\prime}_{\mu}}$, transforms as

[TABLE]

where the identity in eq. (19) is used.

Similar to the non-abelian gauge field $F_{\mu\nu}$, for the isospin field, eq. (40), becomes

[TABLE]

where

[TABLE]

Expanding the star product in power series of $\hbar$ in the zeroth order $\star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial q}\frac{\overrightarrow{\partial}}{\partial p}-\frac{\overleftarrow{\partial}}{\partial p}\frac{\overrightarrow{\partial}}{\partial q}\right)\right]=1$. Then this equation is written as

[TABLE]

Therefore, eq. (48) becomes

[TABLE]

where

[TABLE]

Using eq. (45), this becomes

[TABLE]

The field $\mathbf{B_{\mu}}$ in eq. (47), under a gauge transformation in infinitesimal form, eq. (42), is

[TABLE]

Then the infinitesimal transformation of $\mathbf{B_{\mu}}$ is

[TABLE]

Using $\bm{k}_{\mu\nu}$, the Lagrangian density is written as

[TABLE]

The final Lagrangian density for isotopic spin $1/2$ is

[TABLE]

where $G_{\mu}$ is given by eq. (46) and $\bar{G_{\mu}}$ is

[TABLE]

This result is equivalent to the usual Lagrangian as obtained for Yang-Mills gauge theory. However, for $\epsilon=0$, the covariant derivative reduces, consistently, to the momentum operator in the phase space representation, i.e. $P=p\star$. In the following section these results are used to consider a nucleon in an external field. These results are a test for the consistency of the representation.

IV Nucleon in an External Field

Consider a nucleon in an external field that is intense enough to discard the gauge field self interaction in zero order (natural units are used). Then the Lagrangian in eq. (51) leads to the Dirac equation

[TABLE]

where $\hat{G}_{\mu}$ is given by eq. (46).

Taking $\mathbf{B_{\mu}}=(0,0,B_{\mu})$, then

[TABLE]

with $\alpha=\pm 1$. Except for $\alpha$ this equation is similar to that obtained for the Landau problem in phase space diracphasespace . Writing

[TABLE]

yields $\mathbf{H}=\left(0,0,H\right)$. Then the field equation is given by

[TABLE]

which explicitly reads

[TABLE]

where $\varepsilon=p_{0}$ and $\tau=x_{0}$. For separated variables then two equations are obtained. The first one is given by

[TABLE]

and the second one reads

[TABLE]

Introducing the following variable

[TABLE]

and using $i\sigma^{12}\phi=-s\phi$, with $s=\pm 1$, $\omega=2z/(\alpha\epsilon H)$ and $\phi=\exp{(-\omega)}F(\omega)$, the equation for $F(\omega)$ is

[TABLE]

where $k=\frac{2E^{2}}{\alpha\epsilon}+s$. This equation leads to

[TABLE]

For the negative value of $\alpha$, the following condition holds

[TABLE]

This implies that both states defined by $\alpha$ are physically acceptable. This result expresses the consistency of this representation in phase space. It is important to emphasize that, when a similar result was derived for the electron in an external field diracphasespace , the gauge field in phase space was introduced heuristically. This is not the case here, where the principle of gauge symmetry has led to the derivation of the form for the gauge field.

V Conclusions

A non-commutative-like non-abelian gauge theory is constructed in phase space. This corresponds to a realization of the Seiberg-Witten gauge theory for non-commutative fields that including non-abelian symmetries. The starting point is a Lagrangian density for the free Dirac field in phase space. This leads to a SU(2)- gauge group theory derived in phase space, and associated with the Wigner function. The formalism is applied to study the nucleon in a external isospin gauge field. This corresponds to a Landau problem for isospin, such that the Landau levels are consistently derived for the dublet isospin representation. A similar theory for a generalized non-abelian gauge group will be presented later.

Acknowledgements: This work was partially supported by FAP-DF of Brazil. The work by AES and AFS is supported by CNPq of Brazil.

Appendix A Symplectic field theory

In order to construct a Hilbert space in $C^{\infty}(\Gamma)$, let $\mathcal{H}(\Gamma)$ be a linear subspace of the measurable functions space $\psi:\Gamma\rightarrow\mathbb{C}$ which are square integrable, i.e.

[TABLE]

A Hilbert space is introduced by defining the inner product, $\langle\cdot|\cdot\rangle$, on $\mathcal{H}(\Gamma)$, as $\langle\psi_{1}|\psi_{2}\rangle=\int_{\Gamma}\psi_{1}^{\ast}(q,p)\psi_{2}(q,p)d^{4}pd^{4}q,$ where $(q,p)=(q^{\mu},p^{\mu})$ and $\psi(q,p)$ is defined in $C^{\infty}(\Gamma)$.

Consider the linear mappings $\overline{Q},\overline{P}:\mathcal{H}(\Gamma)\rightarrow\mathcal{H}(\Gamma)$, such that $\overline{Q}\psi(q,p)=q\psi(q,p)$ and $\overline{P}\psi(q,p)=p\psi(q,p)$. Since $[\overline{Q},\overline{P}]=0$, the spectrum of $\overline{Q}$ and $\overline{P}$ is used to construct a basis in $\mathcal{H}(\Gamma)$, $\{|q,p\rangle\}$, such that

[TABLE]

In this case, $\psi(q,p)=\langle q,p|\psi\rangle$, such that

[TABLE]

and $\langle q,p\left|q^{\prime},p^{\prime}\right\rangle=\delta(q-q^{\prime})\delta(p-p^{\prime})$. It is important to emphasize that the operators $\overline{Q}$ and $\overline{P}$ are not the usual quantum mechanics operators for position and momentum. Their physical meaning will be considered latter. For physical interpretation, the state of a system is described by functions $\psi(q,p)$, with the normalization condition

[TABLE]

This symplectic Hilbert space, $\mathcal{H}(\Gamma)$, is taken as the representation space of the Poincaré symmetry. Consider a unitary transformation in $\mathcal{H}(\Gamma)$ which is a linear mapping $U:\mathcal{H}(\Gamma)\rightarrow\mathcal{H}(\Gamma)$, where $\langle\psi_{1}|\psi_{2}\rangle$ is invariant.

Appendix B Abelian gauge theory for the sympletic Dirac field

Here some aspects of the calculation for the abelian gauge-field in phase space are presented. The objective of these results is twofold: first, the work is self contained; and second, many steps (not presented in previous publications Amorim0 ) are important, since these are used in the more intricate case of the non-abelian symmetry as discussed in Section 3.

The Lagrangian density for the Dirac equation in phase space is Amorim0

[TABLE]

This is re-written as

[TABLE]

Our goal is to analyse the invariance of eq. (65) under global and local gauge transformations. First, the analysis is developed in a general way, i.e. without defining the gauge for the system. Using the property of star product, we have

[TABLE]

Then eq.(65) is

[TABLE]

where the star product is introduced. Defining the gauge transformation in the general form leads to

[TABLE]

[TABLE]

In infinitesimal form, eq. (68) takes the form

[TABLE]

where $\psi^{\prime}(q,p)-\psi(q,p)=\delta\psi(q,p)$. Then the gauge transformations, eqs. (68) and (69), are written in infinitesimal form as

[TABLE]

Applying these transformations to eq. (67), leads to the relation

[TABLE]

To facilitate visualization of calculations in eq. (71), the terms in the Lagrangian $\delta\left(p^{\mu}\star\psi(p,q)\right)$ and $\delta\left[\left(\bar{\psi}(p,q)\star p_{\mu}\right)\right]$, are calculated separately. Then the term $\delta\left(p^{\mu}\star\psi(p,q)\right)$ has the form

[TABLE]

Using identities Blaszak

[TABLE]

the term $\delta\left(p^{\mu}\star\psi(p,q)\right)$ is given by

[TABLE]

Using the same procedure for the term $\delta\left(\bar{\psi}(p,q)\star p_{\mu}\right)$ leads to

[TABLE]

Then eq. (71) has the form

[TABLE]

This equation represents a gauge transformation in a general form for the Lagrangian, eq. (67). If the gauge transformation is considered global, where $\lambda$ is a constant, the Lagrangian becomes invariant, i.e. ( $\delta\mathcal{L}=0$ ). However, if $\lambda$ is a spacetime function, i.e , $\lambda=\lambda(q,p)$, a local gauge transformation or gauge transformation of the second kind emerges. An extra term is generated in the above equation $\frac{\partial\lambda}{\partial q^{\mu}}\star\psi(p,q)$, making the Lagrangian non-invariant. To restore gauge invariance a new field, a 4-vector potential $A_{\mu}$ directly connected with $\psi(q,p)$ is introduced. In this case, eq. (73) is rewritten as $\delta\left(p_{\mu}\star\psi(p,q)+\mathcal{L}_{a}\right)$, where the extra term $\mathcal{L}_{a}$ is defined as

[TABLE]

The local gauge transformation leads to

[TABLE]

This equation has extra terms. Two terms are provided by Lagrangian $\mathcal{L}_{a}$, where $(\delta A_{\mu})$ is still unknown, and the other term $\frac{\partial\lambda}{\partial q^{\mu}}$ leads to the term $\left(p_{\mu}\star\psi(p,q)\right)$. In order to cancel the extra term, $(\delta A_{\mu})$ takes the form

[TABLE]

where $e$ represent the charge of the particle, and $\left\{a,b\right\}_{M}=a\star b-b\star a$ is the Moyal bracket. Then defining the operator

[TABLE]

which directly couples to the field $\psi(q,p)$. Applying the gauge transformation of second kind to $D_{\mu}\star\psi(q,p)$, leads to

[TABLE]

Then the operator defined in eq. (79) obeys the covariant transformation rule. Similarly in eq. (74), there is the operator $\delta\left(\bar{\psi}(p,q)\star p_{\mu}+\mathcal{L}_{b}\right)$ where the extra term $\mathcal{L}_{b}$ is defined by

[TABLE]

A local gauge transformation leads to

[TABLE]

Using the definition of $\left(\delta A_{\mu}\right)$ an operator is defined as

[TABLE]

Then the operator, $\bar{\psi}(p,q)\star D_{\mu}$, under a gauge transformation of the second kind is given by

[TABLE]

This satisfies the rule of a covariant transformation. Rewriting the Lagrangian, eq. (67), in terms of operators $D_{\mu}\star$ and $\star D_{\mu}$, leads to

[TABLE]

The new Lagrangian under a local gauge transformation, becomes

[TABLE]

Therefore the Lagrangian described by eq. (82) is invariant under a local gauge transformation. However, the introduction of the field $A_{\mu}$ gives rise to the electromagnetic interaction. To eliminate this interaction, another term is introduced in the Lagrangian for the electromagnetic field. The first step leads to finding the electromagnetic tensor in phase space. Starting with the relation $[D_{\mu}\star,D_{\nu}\star]$, leads to

[TABLE]

For a better understanding, each term is calculated separately. The first term is given as

[TABLE]

The second term in eq. (84) is

[TABLE]

Using the identity, eq.(72), the second term in $T_{4}$ is

[TABLE]

Then the $T_{4}$ term becomes

[TABLE]

By analogy the third term of eq. (84) is written as

[TABLE]

and the last term of eq. (84) becomes

[TABLE]

Therefore the expression for $[D_{\mu}\star,D_{\nu}\star]$ has the form

[TABLE]

Then the electromagnetic tensor is defined as

[TABLE]

where $\mathcal{F}_{\mu\nu}$ is invariant. The invariance of $\mathcal{F}_{\mu\nu}$, is demonstrated using the expression

[TABLE]

Now each term is calculated separately. Using eq. (78) we get

[TABLE]

and

[TABLE]

The last term takes the form

[TABLE]

Then the electromagnetic tensor under local gauge transformation is given as

[TABLE]

Using eq. (66) leads to

[TABLE]

Defining the last term of eq. (82) as

[TABLE]

the final Lagrangian invariant under local gauge transformation has the form

[TABLE]

This demonstrates a mapping similar to that achieved for the Seiberg-Witten gauge for non-commutative fields.

Our goal now is to study the gauge transformation considering the Moyal-Weyl star-product in approximate form, i.e. expanding in power series the star product of general gauge transformation to the zeroth order in $\hbar$. Then eq. (67) is written as

[TABLE]

Expanding the star product of the gauge transformation in the general form, eqs. (68) and (69), in the zeroth order lead to

[TABLE]

Infinitesimal gauge transformation is written as

[TABLE]

Applying this transformation to eq. (87) leads to

[TABLE]

Considering the global gauge transformation of eq. (90) leads to $\delta\mathcal{L}=0$, i.e. an invariant Lagrangian. The Noether theorem gives a conserved current,

[TABLE]

For local gauge transformation, eq. (90) is written as

[TABLE]

Therefore the Lagrangian, eq. (87), is not invariant under local gauge transformation. To restore local gauge invariance, a new four-vector, $A_{\mu}$, coupling directly to the current $J^{\mu}$ is introduced

[TABLE]

where

[TABLE]

and $e$ is the charge of the particle. Using the definition of $(\delta A_{\mu})$, eq. (78), and expanding in power series in $\hbar$ leads to

[TABLE]

Therefore $\delta A_{\mu}$ has the form

[TABLE]

Lagrangian, eq. (92), under a local gauge transformation becomes

[TABLE]

To demonstrate its invariance, ${\cal L}$ is written as

[TABLE]

where the operators are defined as

[TABLE]

and

[TABLE]

Under local gauge transformation, the operator $D_{\mu}$ becomes covariant as

[TABLE]

In order to eliminate the electromagnetic interaction due to the introduction of $A_{\mu}$ power series expansion to zero order in $\hbar$, the star product of eq. (85), is

[TABLE]

The invariance of $F_{\mu\nu}$ is

[TABLE]

where eq. (94) is used. Therefore, the last term of eq. (96) leads to

[TABLE]

and the final Lagrangian is

[TABLE]

The results show a Lagrangian, defined in phase space, is similar to that obtained in quantum field theory.