Commutator derivative. Consider so in a geometric, basis-independent way.

Commutator derivative. We have operators xˆ and pˆ that are clearly somewhat related. The Hamiltonian operator for a quantum mechanical system is represented by the imaginary unit times the partial time derivative. Ask Question Asked 4 years, 4 months ago. Oct 14, 2024 В· Commutator of covariant derivatives and field strength Consider the covariant derivative in some representation of the Lie algebra \\[D_\\mu = \\partial_\\mu - i g A_\\mu^z(x) T_z = \\partial_\\mu -i \\mathbf{A}_\\mu(x). Operators are commonly In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Let each partial derivative operate on everything to the right of it. When the group is a Lie group , the Lie bracket in its Lie algebra is an infinitesimal version of the group commutator. The main examples in field theory have a compact gauge group and we write the symmetry operator as () = where () is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i. We de ne the Lie derivative L A!of !along Aas L A!= d ds May 30, 2020 В· Commutator of covariant derivatives acting on a vector density. General relativity is formulated with the help of a special covariant derivative that is metric-compatible and torsion-free. It is enough for us to deп¬Ѓne the commutator of two vectors by its components in a coordinate basis, [A, µB]=(A ∂ µB ν −Bµ∂ µA ν)e Mar 19, 2021 В· Commutator of covariant derivative for rank 2 tensor. Asking whether $[\dot{},\dot{}]$ is a Lie bracket doesn't really make sense (since, for matrix groups, the Lie bracket is the matrix commutator, anyway). In flat space the order of covariant differentiation makes no difference - as covariant differentiation reduces to partial differentiation -, so the commutator must yield zero. F. Viewed 662 times 0 $\begingroup$ I just started What is the commutator of an operator and its derivative? 2. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. If they were partial derivatives they would commute, but they are not. {\displaystyle A_{\nu ;\rho \sigma }-A_{\nu ;\sigma \rho }=A_{\beta }R^{\beta }{}_{\nu \rho \sigma }. Example \(\PageIndex{1}\) If the operators A and B are scalar operators (such as the position operators) then AB = BA and the commutator is always zero. The commutator of two covariant derivatives, then, measures the difference between parallel transporting the tensor first one way and then the other, versus the opposite. The precise way in which this happens is through exponentiation. The most famous commutation relationship is between the position and momentum operators. Ask Question Asked 3 years, 4 months ago. Curvature tensors are de ned in terms of covariant derivatives. 4 come from. Consider a tensor in some (maybe curvilinear) coordinates \(\,\left\{ x^{i}\right\} \). derivative. Nov 21, 2017 В· $\begingroup$ Your example function is separable and so you just pull the theta out and takes its derivative. Jun 6, 2021 В· The quote you give from Carroll about the covariant derivative is right: it quantifies the rate of change of a tensor field relative to parallel transport. 10, it mentions that in space-time with torsion the commutator of covariant derivatives acting on the Apr 9, 2022 В· Expressions for components of a covariant derivative of a vector and a covector. The gen­eral as­ser­tions are read­ily checked by sim­ply writ­ing out both sides of the equa­tion and com­par­ing. Jun 18, 2017 В· In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyze the gravitational theories, starting from the standard Einstein's general theory of commutator algebra [7, 8]. Sep 21, 2018 В· not fully specify the covariant derivative: we can pick a coordinate system, and choose = 0 in this coordinate system, and de ne a covariant derivative this way { it is just the partial derivative in that coordinate system, which obviously satis es the 3 conditions of covaraint derivatives. We will not use this notation extensively, but you might see it in the literature, so you 9 Operators and Commutators 9. Consider a generic (possibly non-Abelian) gauge transformation acting on a component field = =. Ask Question Asked 8 years, 11 months ago. It is also clear that [ ] = 0 for any operatorAˆ,Aˆ Aˆ. Gauge covariant derivative on form. Based on some new commutator estimates, local existence for the generalized MHD equations is established, which recovers and D. 2. This result extends the well-known commutation relation between one operator and a function of another operator. up to a commutators. del Castillo. Speci cally that the Lie derivative of vector elds X and Y may be thought of as the double derivative of the commutator of their respective ows. In Sec. , a Lie derivative. Consider so in a geometric, basis-independent way. (2. 3 gives their derivation, while Sects. We cannot just recklessly take derivatives of a tensor’s components: partial derivatives of components do not transform as tensors under coordinate transformations. The covariant derivative can be used to construct curvatures (called field strengths in the Yang-Mills case). IV and V present examples of applications to a simple 4-th order operator and the 6-th order operator arising in renormalizable HoЕ™ava gravity theory along with the verification of low-order Gilkey-Seeley coefficients for the heat kernel trace of a generic 4-th order operator. For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +. Let !be a di erential k-form. The Commutator of two operators A, B is the operator C = [A, B] such that C = AB − BA. The physical meaning is simple, that is, transformations on the operative manifold (space-time or particle phase space) due to imposed constraints deп¬Ѓne the "compensation’s. For a function the covariant derivative is a partial derivative so $\nabla_i f = \partial_i f$ but what you obtain is now a vector field, and the covariant derivative, when it acts on a vector field has an extra term: the Christoffel symbol: so that covariant derivatives of tensor elds are still tensor elds. Indeed, given a vector eld V , under a coordinate transformation, the partial derivatives of its components transform as @V 0 @x 0 = @x 6 days ago В· The paper is organized as follows. that the directional derivative can be also de ned by the formula L Af= d ds f As s=0: (1. Each vector п¬Ѓeld X induces a derivation D X just like this. What is the commutator of a reasonably-behaved function of an operator and the derivative of that function? What is $[f(\hat x) ,f'(\hat x) ]$? The commutator of two covariant derivatives, then, measures the difference between parallel transporting the tensor first one way and then the other, versus the opposite ordering. 20. Operators are commonly used to perform a specific mathematical operation on another function. Contents 1 Introduction 3 2 Notation and prerequisites 4 3 The commutator of two vector (a) Prove the following commutator identity: [A, BC] = [A, B]C + B [A, C] . This is the derivation property of the commutator: the commutator with A, that is the object [A,· ], acts like a derivative on the product BC. 5 describes local conformal transformations in the Jul 10, 2019 В· Commutator of functional derivatives. $\endgroup$ – G. Oct 14, 2017 В· Since commutators of covariant derivatives always end up just being some contractions with the curvature, we can immediately see from $(1)$ that the formula will have formalism, including the detailed consideration of what means the covariant derivative of the tetrad. Ask Question Asked 5 years, 3 months ago. Smith. 5. Notation and concepts of Yang Mills Theory. The connection coe cient can then be obtained in any the Lie derivative of a function fis again its directional derivative, L uf= u r f: (7) If u is the 4-velocity of a fluid, generating the fluid trajectories in spacetime, L ufis commonly termed the convective derivative of f. the momentum operator is a derivative in coordinate space and derivatives are related to translations. It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection . If the limits are a function of theta, then the chain rule is required. We would like to know their commutator [xˆ that the directional derivative can be also de ned by the formula L Af= d ds f As s=0: (1. Viewed 72 times 0 $\begingroup$ (this question uses 4 days ago В· The commutator of two group elements and is , and two elements and are said to commute when their commutator is the identity element. If all commutators are vector fields (and all commutators of them are too and etc. Mar 28, 2015 В· Can I use the product rule on the the above expression? More alarmingly the commutator is itself a second derivation! so how come it's a vector field? (obviously it is). The covariant derivative of a tensor at a point doesn't make sense. g. The rescaling by the coupling constant is also convenient In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y]. When you derive a system with respect to two independent variables (which is what the partial derivative does, it ignores your position as a function of time), it doesn't Jan 25, 2014 В· The commutation of derivatives is a mathematical technique used to rearrange the order in which derivatives are applied to a function. However, we need not follow the Cartan notation. Modified 1 year, 11 months ago. with lowered indices) and is Clifford multiplication. I stumbled on Wikipedia's page about interior products (here), and I've noticed a property that so All mixed commutators are zero, since the derivative of one spatial coor-dinate with respect to a different spatial coordinate is zero. If you give me a function and a point, I can take it’s directional derivative at that point in the direction where [,] = is the commutator, is exterior derivative, в™­ = (,) is the dual 1 form corresponding to under the metric (i. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In any case, this is a math question, not a physics one. ) With this choice, vectors become diп¬Ђerential operators (e. This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg). 1) where ais a constant with units of length, making the argument of the exponential unit free. $\endgroup$ – Stack Exchange Network. 20 De­riva­tion of the com­mu­ta­tor rules This note ex­plains where the for­mu­lae of chap­ter 4. of the vector. 71), involving the commutator [X, Y], vanishes when X and Y are taken to be the coordinate basis vector fields (since [,] = 0), which is why this term did not arise when we originally took the commutator of two covariant derivatives. Then L exp(X) p p = Z 1 0 d dt L exp(tX) p dt = Z 1 0 L X ’ t dt = Z 1 0 di X’ t dt = d Z 1 0 i X’ t dt (10) so that a closed p-form and its left translation di er by an exact p-form, and so in particular lie in the same deRham class. Modified 3 years, 4 months ago. The cases of scalar, covariant vector, contravariant vector and arbitrary tensor are considered. The momentum is proportional to the gradient. e. The commutator of two elements, g and h, of a group G, is the element. In probably most cases that one comes across in calculus courses, you can interchange derivative and integral. Mar 15, 2014 В· The derivative of an operator: Let X(t), R → X, where X is some normed linear space, say a Banach or Hilbert space. 3 we describe the construction of the covariant derivative of Dirac fermion. That is, [x;p y]=0 (14) [x;p z]=0 (15) [y;p x]=0 (16) and so on. 0. } The last term in (3. Sec. 2 contains the summary of results, Sect. 4 is deriving the commutator of covariant derivatives and its connection to the Riemann tensor. The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector with itself: [4] [5] A ν ; ρ σ − A ν ; σ ρ = A β R β ν ρ σ . We derive an expression for the commutator of functions of operators with constant commutations relations in terms of the partial derivatives of these functions. $\endgroup$ – AccidentalFourierTransform I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\\hbar$ times the derivative of the position dependent quantity. [g, h] = g−1h−1gh. 1 Operators Operators are commonly used to perform a speciп¬Ѓc mathematical operation on another function. Consider a suitable exponential of the momentum operator: ipa^ e ~; (1. The operation can be to take the derivative or integrate with respect to a particular term, or to multiply, divide, add or subtract a number or term with regards to the initial function. If the Lie group is compact, we can Jan 3, 2020 В· For notation and convention, please see Gauge theory formalism and Generalizing the covariant derivate for gauge theory. Then we can define the derivative in the usual way: ∂tX(t) = lim δ → 0X(t + δ) − X(t) δ. Apr 25, 2018 В· Stack Exchange Network. Scalar field in curved spaces. \\] In the last equation we introduced a Lie-algebra-valued gauge field \\(\\mathbf{A}_\\mu(x) = g A_\\mu^z(x) T_z\\). 2) It turns out that formula (1. 10) If the commutator vanishes, the two operators are said to commute. Jun 18, 2022 В· What is the commutator of an operator and its derivative? 5. It involves swapping the order of differentiation, which can result in different outcomes for the function. Commutator formula between Hessian and Laplacian of a scalar function. We discuss the range of applicability of the formula Jun 4, 2020 В· I am currently looking at "Physical aspect of space-time torsion" by IL Shapiro. Does it make sense to speak in a total derivative of a functional? Part II During one of my daily exercises, I was looking for properties of the elements of Cartan calculus. Jul 14, 2019 В· $\begingroup$ $[u,v]$ is not a commutator per se, it is a Lie bracket, i. As an application, we discuss a class of states, whose squares linearly depend on the states themselves, and give the corresponding Apr 5, 2012 В· Here is a simple proof which I found in the book "Differentiable Manifolds: A Theoretical Phisics Approach" of G. In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. Precisely it is proposition 2. We now want an example for QM operators. However, the commutator of covariant derivatives acting on a point does. Using these results, we can work out the commutators of the angular momentum components with each other. Ask Question Asked 3 years, 7 months ago. The Newtonian limit of u is the 4-vector @ t+ v, and L ufhas as its limit the Newtonian convective derivative (@ Trick with "functional" derivative to evaluate commutators between diagonal hamiltonian and creation fermionic operator 2 If two operators commute, does it mean that every eigenfuction of one is also an eigenfunction of the other? Aug 30, 2019 В· In the previous chapters, we learned how to use the tensor transformation rule. Modified 4 years, 4 months ago. Viewed 451 times is a derivative along di eomorphisms, so is a Lie derivative. If one knows the components of the tensor in these coordinates and the relation between original and new coordinates, \(x^{\prime i} = x^{\prime i} \left( x^{j}\right) \), it is possible to derive the components of the tensor in May 2, 2021 В· This paper considers the problem of the local existence for the generalized MHD equations with fractional dissipative terms $$\\Lambda ^{2\\alpha } u$$ Λ 2 α u for the velocity field and $$\\Lambda ^{2\\beta } b$$ Λ 2 β b for the magnetic field, respectively. . Modified 2 years, 2 months ago. A = Aµ∂µ)and thus the commutator of two vector п¬Ѓelds involves derivatives. The Bianchi identity states that the covariant derivative of the Riemann curvature tensor is equal to the commutator of the covariant derivatives of the metric tensor. ) this means that all these higher derivatives are actually first order derivatives? What's Mar 15, 2016 В· To show this for vector fields, we use the relation $$ \mathcal L_XY=[X,Y], $$ and hence the statement follows from the properties of the Lie bracket of vector fields. Nov 7, 2020 В· Put a field, say a scalar $\phi$, after the commutator. Sect. This approach is founded on principles concerning various orders of covariant derivatives’ commutators. There in eq. The actual computation is very straightforward. We’ll do one and use cyclic per- Sep 5, 2013 В· The covariant derivative of a commutator is directly related to the Bianchi identity, which is a fundamental equation in differential geometry. T. This method is originated from the Lyapunov representation and would be very useful for cases that the anti-commutators among the state and its partial derivative exhibits periodic properties. Viewed 3k times 2 $\begingroup$ I am a Jul 9, 2020 В· In this work we find expression for commutator of covariant derivative and Lie derivative. 2) can be generalized to de ne an analog of directional derivatives for di erential forms and vector elds, which is the Lie derivative. In the result the commutator is п¬Ѓrst taken with B and then taken with C while the operator These examples show that commutators are not specific of quantum mechanics but can be found in everyday life. Commutators are Lie brackets, in this case on the algebra of differential operators. Can someone point me towards a commutator [A,B] of two operators, deп¬Ѓned to be the linear operator [Aˆ, Bˆ ] ≡ AˆBˆ − BˆAˆ. Commuting the derivatives reverses the order of operations, but unless your space Commutator of covariant derivatives. We de ne the Lie derivative L A!of !along Aas L A!= d ds Jun 17, 2021 В· Commutator of Lie derivative. rfuwuyl imxeq rqigo awv fwljhk qgcg sht kiogwg sav nlz