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Everything posted by Linker

  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.
  16. Every time I'm asking, why almost every woman have almost no mental srengths; they have big problems to be courageous, there are almost no women in military institutions, almost no female Heavy Metal bands, etc. Moreover, women compared with men are physically weak. I have set my thoughts here: http://linkerdynamics.wordpress.com/2014/05/09/why-men-are-the-better-soldiers-and-women-are-the-better-nurturers/ But now I offer my question: What are the molecular, not evolutionary reasons for that women feel fear more intense or why there is a "Tend or be friend" syndrome in women when they are under stress? I search for answers for example like this: "Women feel fear more intense because they have statistically more genes that are coding for enzymes that set the brain structure so that there is more activity in Amygdala. Furthermore because they have less testosterone and more estrogen the stress reaction is regulated in that way that there is released more oxytocin; Oxytocin regulate that genes that are required for social interactions, crying with tears, etc..."
  17. Yes, I am asking why in general relativity case the covariant divergence of the energy-momentum tensor hold, too. Assuming, a stone is falling down on earth on the time point T and given its trajectory x(T). Repeat the same initial conditions (stone has same mass, same shape, there are same environmental conditions, etc.) on time point T+DeltaT. Then x(T+dT)=x(T) even when there takes gravity place. Or I say it better, if there is Newtonian gravity. Treating this phenomenon with General Relativity, the stone must increase its mass (because it accelerates and gains kinetic energy which is extra mass by special relativity) and there are some deviations from Newton's theory. Can be really assumed x(T+dT)=x(T) in General Relativity case or is energy-momentum not really conserved by treating phenomena with General Relativity?
  18. Why the laws of gravity are space-time-translation invariant, too? I don't know what is the fundamental reason fore space-time symmetry...
  19. May be A, B initial conditions and C a phenomenon that occur if A and B are satisfied. Now, consider A' and B' as the same initial conditions, but translated arbritary in space and time. Then, these initial conditions (A' and B') will imply C' the same phenomenon C but translated arbritary in space and time. Is there a reason why every Lagrangian is space/time translation invariant? General relativity (and maybe Quantum gravity) are space-time-translation invariant theories, too, because the laws of gravity are space/time independent, right? How does a quantum field theory in curved background look like?
  20. Hi, in classical physics, in a closed system energy is conserved for every instant of time. But in quantum physics, quantum fluctuations (very short particle/energy creation or annihilation) on microscopic level can occur. But both theories have the same propertiy that the Noether theorem holds: If the theory is invariant under time translations the energy is conserved. In every quantum field theory there is time translation invariance. But are there theories that are time dependent (i.e. theories that break homogenity in time)? Why, the homogenity in time and homogenity in space are the most essential symmetries in physics (of course there are theories that violate symmetries, e.g. CPT-symmetry)?
  21. Hello, I don't believe that materializations of spirits is really existent, but I have seen in the internet something about materializations: http://en.wikipedia.org/wiki/Albert_von_Schrenck-Notzing If you look at the picture of Eva C. you will think that the picture is faked... But now a question on physics: From quantum field theory/ and particle physics it is well-known that particles can be created from vacuum for a short time. But a human being consist of billions of particles. So if someone starts to compute Feynman diagrams with billions of elementary particles and (using standard model of particle physics) it will be an enormous expenditure; but the probability of having billions of Loop contributions in a Feynman diagram tends to zero. So, from quantum field theory materializations of macroscopic objects is zero (except there is a extremely magnificient energy that enables to create many particles). Naturally, on our earth there is not such a large amount of energy that could create many particles from vacuum for a while. Which physical requirements must hold that macroscopic objects can be materialized from vacuum? What would happen, if Planck's constant would attain a much larger value? Is Planck's constant really an universal constant of nature or can it change under certain requirements?
  22. Einstein's GR is a theory without torsion. A gravitational theory with torsion is the Einstein-Cartan-theory. This theory predicts that a particle with spin causing torqued spacetime, but only on microscopic scales. What would happen if the spacetime is torqued on macroscopic scales? Is it possible to induce torsion of spacetime on macroscopic scales? If yes, how it can be realized?
  23. Now I have a better idea. The spacetime is characterized by algebraic expressions. Assuming a polynomial function [latex]f(x)= (x-a_1)(x-a_2)...(x-a_n)[/latex] where the zeroes are given by [latex]a_k = r_k e^{\frac{2 \pi i}{k}}[/latex]. With increasing k the zeroes fill a closed curve in the complex plane. And this closed curve can be extended to the 4-dimensional spacetime. For the metric tensor, the following relation holds: [latex]g_{\mu \nu} = \frac{\partial \vec r}{\partial x^\mu}\frac{\partial \vec r}{\partial x^\nu}=|\frac{\partial \vec r}{\partial x^\mu}||\frac{\partial \vec r}{\partial x^\nu}|cos(\frac{\partial \vec r}{\partial x^\mu},\frac{\partial \vec r}{\partial x^\nu})[/latex]. So, there can be defined a vector potential [latex]g_\mu:=|\frac{\partial \vec r}{\partial x^\mu}|[/latex] and the cosine between two vectors can be quantized by the zeroes of polynomials. Because the spacetime geometry is not determined by energy-momentum-tensor as in GR (all shapes of spacetime are allowed but the shapes have quantum-mechanical probabilities), the probability for any angle between two vectors is given by the probability that the vectors lying on the zeroes that have this angle difference. Algebraic theories like the Galois theory can be used to define quantizations of angles. For the quantized vector field [latex]g_\mu[/latex] (the field of characteristic lengths) there can be used the relationship for vector calculus [latex]\frac{\partial \vec r}{\partial x^\mu}=g_\mu cos(\frac{\partial \vec r}{\partial x^\mu},\vec e_\alpha) \vec e^\alpha[/latex] The quantized metric tensor is invariant under the rotation of the zeroes in the polynomial function, i.e. the transformation [latex]\frac{\partial \vec r}{\partial x^\mu} \mapsto T \frac{\partial \vec r}{\partial x^\mu}[/latex] with the orthogonal transformation matrix T. All other quantities like Christoffel's symbols, Riemann's tensor, etc. can be derived from the characteristic lengths and the angles between them. The action functional must be invariant under the rotation of the zeroes. For the path integral quantization, there must be integrated over - the vector potential [latex]g_\mu[/latex] - all possible algebraic realizations of polynomials - this algebraic realizations including the angles lying between two characteristic lengths
  24. My theory is the following: The metric tensor [latex]g_{\mu \nu}[/latex] can be expressed in dependence on a 4x4-matrix field [latex]\omega[/latex] with some algebraic classifications (my matrices are classified by the degree of commutativity). So, the metric tensor can be expressed as [latex]g_{\mu \nu} =e_\mu \omega e_\nu + O(\omega \omega)[/latex] with characteristic unit vecors [latex]e_\mu[/latex]. The degree of commutativity is the following: May be [latex]A,B[/latex] two matrices. Then, the entries of that matrices determine the commutator result [latex]C:=AB-BA[/latex]. Different commutator results are possible: The matrices can commute ([latex]C=0[/latex]) or anticommute ([latex]C= 2AB[/latex] or [latex]C=-2AB[/latex]), etc. The matrix norm of the commutator is a possible variable for the degree of commutativity. Furthermore, the four eigenvalues of the matrix are gravitation field variables. Because every matrix can be written as [latex]\omega = Tdiag(\lambda_1,...,\lambda_4)T^{-1}[/latex], other field variables are algebraic variables depending on the structure of the diagonalization matrix [latex]T[/latex]. The action functional of my theory has the form: [latex]S=\int d^4x \sqrt{-g(\omega)}(g^{\mu \nu}(\omega)tr(\partial_\mu \omega^\dagger \partial_\nu \omega) + L_{source}(\omega))[/latex] But that's not all. The path integral function has the form: [latex]K = \int d \lambda_1... \int d \lambda_4 \oint d [a] e^{iS} [/latex] Hence, the integration over all algebraic possibilities [latex][a][/latex] makes the theory so complicated. In General Relativity, the gravitational field couples on the 4x4 = 16 functions of the energy-momentum-tensor. In this quantum gravity theory the coupling is similar to GR (remind that the noncommutative algebra of matrices caused by its non-diagonal elements), but at first the complicated algebra integration must be performed. With the approximation [latex]\oint d [a] e^{iS} \approx \sum_{[a]}e^{iS} > (\prod_{[a]} e^{iS})^{\frac{1}{N_{[a]}}} = exp(\frac{i}{N_{[a]}}\sum_{[a]} S) [/latex] ([latex]N_{[a]} [/latex] is the effective number of algebraic possibilities) there can be approximated the GR Lagrangian. By applying the Gaussian law in the Lagrangian with vanishing boundary terms [latex]S=\int d^4x \sqrt{-g(\omega)}g^{\mu \nu}(\omega)tr(\partial_\mu \omega^\dagger \partial_\nu \omega) = \int dS_\mu (...)^\mu - \int d^4x \sqrt{-g(\omega)}g^{\mu \nu}(\omega)tr(\omega^\dagger \partial_\mu \partial_\nu \omega) - \int d^4x \sqrt{-g(\omega)}\partial_\mu (g^{\mu \nu})(\omega)tr(\omega^\dagger \partial_\nu \omega) [/latex] the action principle contains the elements of the linear part of the ricci tensor in the second term and the elements of the nonlinearities in the third term. Note that not the KG equation, but the sum over [latex][a][/latex] creates the structure of the ricci tensor.
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