New gauge theory - is it plausible?

[Linker]

I am thinking about a new type of quantum gauge theory: tensor product algebra gauge theories. The tensor product algebra should not be mixed up with a tensor product algebra in mathematical sense: I am assuming that the tensor product algebra can be embedded into a vector space with global basis [latex]\xi_\sigma[/latex]. The fermion field can be expressed as:

[latex]\psi = \psi_A e^A + \psi_{AB} e^A \otimes e^B + \psi_{ABC} e^A \otimes e^B \otimes e^C = \psi_\sigma \xi^\sigma[/latex]

Here, [latex]e^A[/latex] is the "untensored" basis which depends on spacetime and [latex]\psi_A, \psi_{AB}, ...[/latex] are also spacetime dependent spinor functions. Every tensorization of [latex]e^A[/latex] leads to a new basis element due to embedding of tensor products into the vector space. A reciprocal basis [latex]\xi_\sigma[/latex] can be constructed.

 

Now I define a gauge transform: The constructed infinite-dimensional vector space [latex]\xi_\sigma[/latex] can be "rotated" to another [latex](\xi_\sigma)'[/latex] with the local gauge transformation matrix [latex]g^\lambda_\sigma[/latex]. Thus, there must exist a gauge covariant derivative

[latex](D^\lambda_\sigma)_\mu[/latex] where the connection [latex](\Gamma^\lambda_\sigma)_\mu[/latex] transforms under gauges very similar to any other gauge connection in quantum field theory. However, due to the tensor product algebra structure "sitting" in this infinite-dimensional vector space, the gauge connection has another behavior than well-known gauge connections. Assuming that the change of the basis element [latex]e^A \otimes e^B[/latex] with space-time is only a linear combination of elements [latex]e^A, e^A \otimes e^B[/latex] and not dependent on higher tensored basis elements (when describing structure of a simple tensor one do not has to deal with more complicated tensors). Then the corresponding gauge connection matrix reads

[latex](\Gamma^\lambda_\sigma)_\mu \xi_\lambda \otimes \xi^\sigma[/latex]

where the basis elements [latex]\xi^\sigma[/latex] must be the same or lower tensored elements than [latex]\xi_\lambda[/latex]. This restriction makes a difference to ordinary Yang-Mills-theories. Therefore, a more general field strength tensor can be obtained.

 

How I came to this gauge theory: I have thought about interacting and independent collections of particles. When describing a complicated particle system with a quantum field, I have assumed that there are diffenent kind of indepenent particle states. I have also assumed that one single tensored state could also be expressed as a linear combination of different tensored states (gauge symmetry).

 

Question: Is my theory plausible? Can it be regarded as a generalization of Yang-Mills theories?

Edited July 22, 2014 by Linker

[ajb]

I don't know if it is related to what you are trying to do, by the notion of a Quillen superconnection springs to mind. Here instead of a connection one-form you have an inhomogeneous form which must be Grassmann odd. I wonder if (though there is now a question of symmetric vs antisymmetric) you fermion can be understood in this way.

 

As you have presented it your theory is classical right? You need to construct an action and see if this can be quantised.

Edited July 22, 2014 by ajb

[Linker]

I don't know if it is related to what you are trying to do, by the notion of a Quillen superconnection springs to mind. Here instead of a connection one-form you have an inhomogeneous form which must be Grassmann odd. I wonder if (though there is now a question of symmetric vs antisymmetric) you fermion can be understood in this way.

 

As you have presented it your theory is classical right? You need to construct an action and see if this can be quantised.

 

The vector space basis can be represented as follows:

 

[latex]\psi = \sum_{i=1}^\infty (\prod_{j=1}^i \sum_{A_j = 1}^N)_{A_1 \geq A_2 \geq ... \geq A_i} \psi_{A_1...A_i}e^{A_1...A_i} := \psi_{\sigma} \xi^{\sigma}[/latex]

 

Here, [latex]e^{A_1...A_i}[/latex] is the basis element corresponding to the quasi-tensorproduct [latex]e^{A_1} \otimes ... \otimes e^{A_i}[/latex] and N is the dimension of the "untensored" basis vectors. I have found out that my theory works only if the tensor products are symmetric (I require a symmetric algebra). Therefore, index labeling has to be ordered [latex]A_1 \geq A_2 \geq ... \geq A_i[/latex].

 

Now I define the connection 1-form ([latex]\mu[/latex] is the Minkowski space-time index and [latex]\sigma, \lambda[/latex] are running over the complete vector space basis):

[latex]\Gamma_\sigma^\lambda = (\Gamma_\sigma^\lambda)_\mu dx^\mu[/latex].

 

When [latex]d[/latex] is the exterior derivative operator, it holds the connection equation:

[latex]de^{A_1...A_n} = \sum_{i=1}^n (\prod_{j=1}^i \sum_{B_j = 1}^N)_{B_1 \geq B_2 \geq ... \geq B_i} e^{B_1 ... B_i} (\Gamma_{B_1 ... B_i}^{A_1 ... A_n})_\mu dx^\mu[/latex].

 

(Can also be written by using the exterior covariant differential operator D as [latex]De^{A_1...A_n} = 0[/latex] for all n)

 

When the derived basis element has tensorial degree n, its spacetime change is given by linear combinations of basis elements of equal or lower degree. The B-indexed basis must have therefore the same or lower degree than the A-indexed basis.

 

The action of my theory has the following form

 

[latex]S = \int d^4x [i \psi_\lambda^\dagger \gamma^\mu (\delta_\sigma^\lambda \partial_\mu + (\Gamma_\sigma^\lambda)_\mu) \psi^\sigma + \alpha (F_\sigma^\lambda)_{\mu \nu}(F_\lambda^\sigma)^{\mu \nu}] + S_{gf}[/latex]

 

with coupling constant [latex]\alpha[/latex] and 2-form field strength tensor [latex]F_\lambda^\sigma = (F_\lambda^\sigma)^{\mu \nu} dx_\mu \wedge dx_\nu[/latex]. Looks similar to ordinary gauge quantum field theories (my theory should consist also a gauge fixing contribution with Fadeev-Popov ghosts). The difference between a Yang-Mills theory and my theory arises from the assumptions above (the gauge connection is different from other theories). I have defined the link between connection and field strength tensor as:

[latex]F_\lambda^\sigma \xi_\sigma \otimes \xi^\lambda = D \Gamma^\sigma_\lambda \xi_\sigma \otimes \xi^\lambda[/latex]

Due to [latex]deg(\lambda) \geq deg(\sigma)[/latex], covariant derivatives are different from ordinary covariant derivatives.

 

For quantization Feynman's propagator can be used. Is my theory plausible?

[ajb]

I guess you should carefully show that you really do have a connection. It all transforms in the right way? Geometrically where does this conenction live? (Maybe a side question really).

 

You should try applying gauge fixing and the FP-ghosts etc and see it all works. You may have to think about local anomalies and so on. It has been a while since I looked at these things in any detail.

 

By plausable I guess you mean "is it as well defined as any classical/QFT is?" I don't know, you need to work out some details.

 

I am trying to think if I have seen anything similar anywhere. If I think of something I will let you know.

[Linker]

The connection lives in a vector space [latex]V = span(\xi_\sigma)[/latex]. More formally, V is the space [latex]V = \Pi(T^\infty E/(E_1 \otimes E_2 - E_2 \otimes E_1))[/latex], where [latex]T^\infty E[/latex] denotes the tensor product algebra over reference vector space E and [latex]\Pi[/latex] is the projection of any tensored element to a vector space element. The set of projections (linear operators) consists of all elements [latex]\pi_{21}, \pi_{32}, \pi_{43}, ... \in \Pi[/latex], where [latex]\pi_{i,i-1}[/latex] projects a tensor product of i-th degree on a tensor product of degree i-1. It holds [latex]\pi_{i,i-1} = id[/latex] when applied to an untensored (tensor product of degree 1) element. The fiber bundle that describes the gauge theory has the form [latex]U := M \times \Pi((T^\infty E)_{sym})[/latex] with the minkowski spacetime M. Every element (section) of this fiber bundle can be represented as:

[latex]\prod_{i=1}^\infty \pi_{i+1,i}(x_\mu) \sum_{j=1}^\infty (c_{A_1 ... A_j}e^{A_1} \otimes ... \otimes e^{A_j})(x_\mu) \in U\[/latex].

 

This mathematical concept also leads to the connection mentioned above: The element [latex]e^{A_1 ... A_n} = \prod_{i=1}^\infty \pi_{i+1,i}(x_\mu)e^{A_1} \otimes ... \otimes e^{A_n} = \prod_{i=1}^{n-1} \pi_{i+1,i}(x_\mu)e^{A_1} \otimes ... \otimes e^{A_n}[/latex] has the exterior derivative:

[latex]de^{A_1 ... A_n} = \sum_{i=1}^{n-1} \pi_{n,n-1} ... d \pi_{i+1,i} ... \pi_{21}e^{A_1} \otimes ... \otimes e^{A_n} + \prod_{i=1}^\infty \pi_{i+1,i}d (e^{A_1} \otimes ... \otimes e^{A_n})

[/latex]

The last term is the ordinary tensor product derivative by the use of the Leibnitz rule and all other terms are the derivatives of projections. With the assumption that the projection operator transforms similar to the reference vector space elements [latex]d \pi_{i+1,i} = \Gamma \pi_{i+1,i}[/latex] with specific connection operator [latex]\Gamma[/latex] above connection can be obtained.

 

The rest is similar to all other quantum gauge theories: Gauge transform matrix is [latex]g_\sigma^\lambda = (exp(ih))_\sigma^\lambda[/latex] with hermitian matrix [latex]h[/latex] so that the connection [latex]\Gamma_\sigma^\lambda e^\sigma \otimes e_\lambda[/latex] transforms to

[latex](\Gamma_\sigma^\lambda + idh_\sigma^\lambda) g^\sigma_\kappa e^\kappa \otimes g_\lambda^\eta e_\eta[/latex] by applying this gauge transform. Hence, [latex]d\Gamma_\sigma^\lambda e^\sigma \otimes e_\lambda[/latex] transforms to [latex]d\Gamma_\sigma^\lambda g^\sigma_\kappa e^\kappa \otimes g_\lambda^\eta e_\eta[/latex] so that this 2-form field can be assumed as the field strength tensor. When defining a suitable symmetry group, one can represent the matrix h in terms of generators T, i.e. [latex]h_\sigma^\lambda = h_a(T^a)_\sigma^\lambda[/latex] and so it is clear how to represent any other matrix-like components in a basis of generators.

 

Gauge fixing: I use the usual gauge fixing condition [latex]\partial^\mu ((\Gamma_\sigma^\lambda)_\mu e^\sigma \otimes e_\lambda) = k_\sigma^\lambda e^\sigma \otimes e_\lambda[/latex] for an arbitrary matrix-like scalar field function k. When an infinitesimal gauge transform is performed, the matrix-like scalar expression k reads [latex]\partial^\mu (((\Gamma_\sigma^\lambda)_\mu + i\partial_\mu h_\sigma^\lambda) (e^\sigma + ih_\kappa^\sigma e^\kappa) \otimes (e_\lambda + ih_\lambda^\eta e_\eta))[/latex] when h is infinitesimally small. When expressing matrices in terms of generators (and finding a link between commutators or some more complicated mathematical expressions of generators and a linear combination of generators) the Fadeev-Popov determinant can be obtained by deriving by [latex]h^a[/latex] and using Grassmann-valued FP-Ghosts. The gauge fixing term can be obtained by integrating over k on both sides of the Feynman propagator.

 

Am I right?

Edited July 23, 2014 by Linker

[ajb]

I am not sure, let me think about it (if I get chance!)

[Linker]

I am not sure, let me think about it (if I get chance!)

 

Sure, you can think abute it!

 

Hint: For the generator representation I use an ascending generator basis. This is given as follows:

A matrix that maps from [latex]e^A[/latex] to [latex]e^B[/latex] has the generator frame [latex]span(T^a)[/latex].

When a matrix maps an element [latex]e^A[/latex] to [latex]\pi_{21}(e^{B_1} \otimes e^{B_2})[/latex] this matrix has the extended generator frame [latex]span(T^a,T'^a)[/latex].

For a map from [latex]e^A[/latex] to [latex]\pi_{32} \pi_{21}(e^{B_1} \otimes e^{B_2} \otimes e^{B_3})[/latex] the generator basis is also extended; the frame is then [latex]span(T^a,T'^a, T''^a)[/latex].

etc. ...

The use of an ascending basis satisfies the connection equation and allows to use a structurized framework for the generators of the symmetry group (for example there might be symmetries in and between degree 2 tensor basises and degree 3 tensor basises but other basises have no symmetries).