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Quark

Quark (2/13)

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  1. 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).
  2. 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?
  3. 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?
  4. 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?
  5. Does the average behavior of a population doesn't change over time?
  6. It seems that fundamental physical laws like energy and mometum conservation hold for every time. Even biological phenomena (that are basing on a couple of physical processes like ion transport, etc.) are present every time. It seems that an organism is preprogrammed by its DNA and circulating hormones. (The environment can affect also the organism because the organism is an open system.) What are the reasons that laws of nature doesn't change (every process in the universe is reproducable)?
  7. For the people who are familar with continuum mechanics: How I can derive a stress tensor T for a chocolate dependent on - Cauchy-Green deformation tensor E - Strain rate tensor D (is objective derivative of E) - The time where the chocolate is exposed mechanical loads or temperatures (denoted by t) - The state of the chocolate (whether it is solid or liquid) - etc.? I am very familar with continuum mechanics, I know how to derive material models for viscoelastic, plastic or viscoplastic continua. But chocolate seems to be more complicated than these, because chocolate will undergo a phase change if it is chewed in the mouth. I have to use diffusion equations also for deriving the stress tensor, right? Do I need in-between configurations for describing chocolate in different phase states? Are there theories which can describe such complicated material behavior?
  8. I have seen some computer animations where collisions of motorbikes are simulated. These animations are really realistic; even the dynamics of the driver is visualized exactly. When a motorbike with a given velocity collides with an obstacle, the driver flies off the motorbike and falls to the ground. Why the motorbike driver flies a certain way and why he can spin around in the air? On which equations foundate computer animation engines that simulates such behavior? What is the most hazardous factor injuring a motorbike driver? It is the speed from the motorbike or the falling on the ground? From which factors the lethality of a motorbike accident depends?
  9. The condition (i) reads: [latex]a^{* \mu}a_{\mu}=1[/latex]. Now, [latex]a^{* \mu}[/latex] is a gauge field (like the photon field). Its gauge transformation is given by: [latex]a^{* \mu} \rightarrow a^{* \mu}+ a^{* \nu} \partial^{\mu} a_{\nu}[/latex]. Inserting this gauge transformation into condition (i) and requiring gauge fixed one obtains condition (ii) The field [latex]a_{\nu}[/latex] is the generating field for the gauge. What I'm meaning with generating field: For a photon it is the field [latex]\phi[/latex] with gauge transformation [latex]A_\mu = A_\mu + \partial_\mu \phi[/latex]. Now I have to vary with respect to [latex]a_{\nu}[/latex] (because this is the generating field) when computing [latex]J[/latex]. But how I gain the Faddev-Popov determinant in this case?
  10. This topic is not a question about a quantum field theory; it is about how to express [latex]J:= \delta(a^{* \mu}a^{\nu} \partial_{\nu} a_{\mu})[/latex] in terms of arbritary functions on the spacetime [latex]a_{\mu},a^*_{\mu}[/latex]. There is no need to know in what sense these vector fields are used. What I should do when I express [latex]J[/latex] in terms of Fadeev-Popov ghost fields?
  11. Assuming that [latex]a_{\mu}[/latex] is a gauge function and [latex]a^*_{\mu}[/latex] is the gauge field, then the generalized rule for delta distribution is (applied for condition (ii)): [latex]\delta(a^{* \mu}a^{\nu} \partial_{\nu} a_{\mu}) = \int da'^{\nu} \frac{ \delta(a^{\nu}-a'^{\nu})}{det(Da^{* \mu}a^{\nu} \partial_{\nu} a_{\mu})}[/latex] Here, [latex]a'^{\mu}[/latex] are all the [latex]a^{\mu}[/latex] that obeye condition (ii). The Jacobian for condition (ii) is formed by the operator D that means D varies condition (ii) functionally by [latex]a^{\mu}[/latex]. Am I right from mathematics? When condition (ii) is varied, one obtains [latex]det(Da^{* \mu}a^{\nu} \partial_{\nu} a_{\mu}) = det((a^{* \beta} \partial^{\nu} a_{\beta} - a^{* \mu}a^{\nu} \partial_{\nu})Da_{\mu})[/latex]. How I should proceed?
  12. Hello, when I have a quantum field theory with action S(psi, a_mu, a*_nu) dependent on fermion field psi and dependent on the two boson fields a_mu and a*_nu. All boson fields are complex numbers; no noncommutative objects like matrices. These boson fields are connected with two conditions. In the whole theory there should be satisfied the following conditions: (i) a_{mu}a*^{mu}=1 (ii) a^{mu} a*^{nu} d_{mu} a_{nu} = 0 Here, d_{mu} is the ordinary partial derivative in spacetime direction mu. The Feynman propagator reads: f = e^(iS) delta(a_{mu}a*^{mu}-1) delta(a^{mu} a*^{nu} d_{mu} a_{nu}) (Integration over all psi, a_mu and a*_nu) The first delta distribution (condition (i)) can be written as an integral over e^(i integral d^4x K(x) (a_{mu}a*^{mu}-1))/(2 pi) by the variable K(x). But how I can express condition (ii) in the propagator? Condition (ii) contains a spacetime derivative. How I can transform delta(a^{mu} a*^{nu} d_{mu} a_{nu}) in the Feynman propagator? Shall I use faddeev-popov ghosts (despite it is a commutative theory)?
  13. How testosterone or other sex hormones affects the stress reaction? Is there a molecular mechanism for testosterone on the Amygdala region?
  14. I think so: 1 year old girls are more fear sensitive then 1 year old boys (Literary source: Pink brain, blue brain - Author: Lise Elliot). Lise Elliot thinks that the sex differences in fear and risk taking bases on neuronal activity differences in male and female brains. Women percept fear with the right half of the Amygdala (the brain region for emotions), men percept fear with the left half of the Amygdala. So, when projecting these research facts on everyday life, right Amygdala has the neural circuits for the "Tend and be friend" syndrome. Moreover, there are not so much social influences that make women less riskier and more emotional. Only the parents that trust more in physical abilities of boys rather than girls or some informations from the environment. In my eyes it seems that women are generally weak. The case that any woman is interested in a military job is very rare. "The more testosterone a person has, the more it is interested in a military job", so I can think. What are the biochemical reactions for a stress reaction that is affected by testosterone/estrogen? Are there any biochemical evidence that women are more emotional and what are the molecular mechanisms?
  15. I think that women are very sensitive with their emotions, because I have seen and heard it very often.
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