Functional/variational derivative and the Leibniz rule - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:10:47Z http://mathoverflow.net/feeds/question/109717 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109717/functional-variational-derivative-and-the-leibniz-rule Functional/variational derivative and the Leibniz rule Sietse Ringers 2012-10-15T13:29:39Z 2012-10-15T16:45:50Z <p>I am currently trying to understand the BV-formalism, which makes heavy use of the functional derivative.</p> <p>Let us consider the <em>functional derivative</em>, as defined in for example <a href="https://en.wikipedia.org/wiki/Functional_derivative" rel="nofollow">its Wikipedia article</a>. Let $F$ be a functional, i.e. a map from, say, $C^\infty(\mathbb{R})$ to $\mathbb{R}$, and suppose it may be written as $F[\phi] = \int f\big(x,\phi(x),\phi'(x),\dots,\phi^{(n)}(x)\big)\,dx$ for some function $f$ which depends on the derivatives of $\phi$ up to order $n$. Then the functional derivative of $F$ is $\displaystyle \frac{\delta F}{\delta \phi} = \sum_{i=1}^n(-1)^i\frac{d^i}{dx^i}\frac{\partial f}{\partial \phi^{(i)}}$.</p> <p>Now, my background is that of differential equations and differential geometry, i.e. jet spaces and variational calculus and the like. In that area, the latter operator, $\sum_{i}(-1)^i\frac{d^i}{dx^i}\frac{\partial}{\partial \phi^{(i)}}$, is well known; it is called the <em>variational derivative</em>. Summarizing, then, we seem to have that the <em>functional</em> derivative of a functional is the <em>variational</em> derivative of (one of its) densities.</p> <p>Since the variational derivative involves lots of derivatives, it certainly does not satisfy the Leibniz rule, i.e. it is not a derivation. In various places, however, I've come across the statement that the functional derivative <em>does</em> satisfy the Leibniz rule. (That already seems unexpected to me: how can an operator which is so intimately connected to a decidedly non-derivation be a derivation?) There are various ways to prove it, but I would like to understand this fact in terms of the variational derivative, if possible. So: how can the Leibniz rule of the <em>functional</em> derivative related to <em>variational</em> derivative; can the former be expressed somehow in terms of the latter?</p> http://mathoverflow.net/questions/109717/functional-variational-derivative-and-the-leibniz-rule/109731#109731 Answer by Igor Khavkine for Functional/variational derivative and the Leibniz rule Igor Khavkine 2012-10-15T15:34:37Z 2012-10-15T16:45:50Z <p>Connection of functional derivative with variational derivative: $\frac{\delta}{\delta\phi(x)} F[\phi] = \frac{\delta F[\phi]}{\delta\phi}(x)$. Note that the variational derivative carries an extra coordinate variable dependence. It helps to make it explicit when there is similar confusion.</p> <p>Functional derivative Leibniz rule: $\frac{\delta}{\delta\phi(x)} F[\phi] G[\phi] = \frac{\delta F[\phi]}{\delta\phi}(x) G[\phi] + F[\phi] \frac{\delta G[\phi]}{\delta\phi}(x)$.</p> <p>Special case: $F_x[\phi] = \phi(x)$, $G_{i,y}[\phi] = (\partial_i\phi)(y)$, and $$\frac{\delta}{\delta\phi(z)} F_x[\phi] G_{i,y}[\phi] = \delta(x-z) (\partial_i\phi)(y) - \phi(x) \frac{d}{dz_i}\delta(y-z)$$.</p> <p>Notice the distributional coefficients in the derivatives. There is no way to get away from them if you wish to consider $\phi(x)$ and such as functionals in their own right.</p> <p>If you are interested in the BV formalism in the physics formalism, where the distinction between the functional and variational derivatives is barely remarked, I recommend the reviews by Henneaux and by Gomis, París and Samuel: <a href="http://dx.doi.org/10.1016/0920-5632%2890%2990647-D" rel="nofollow">doi:10.1016/0920-5632(90)90647-D</a>, <a href="http://dx.doi.org/10.1016/0370-1573%2894%2900112-G" rel="nofollow">doi:10.1016/0370-1573(94)00112-G</a>. If you are interested in the BV formalism purely from the point of view of jets, without bringing functionals into the picture, other than peripherally, I recommend the early paper of McCloud and this sequence of papers by Barnich, Brandt and Henneaux: <a href="http://arxiv.org/abs/hep-th/9307022" rel="nofollow">arXiv:hep-th/9307022</a>, <a href="http://arxiv.org/abs/hep-th/9405109" rel="nofollow">arXiv:hep-th/9405109</a>, <a href="http://arxiv.org/abs/hep-th/9405194" rel="nofollow">arXiv:hep-th/9405194</a>, <a href="http://arxiv.org/abs/hep-th/0002245" rel="nofollow">arXiv:hep-th/0002245</a>. If you are more interested in the BV formalism more from the functional point of view, with the appropriate level of functional analysis included, and with jets appearing only peripherally, I recommend the papers by Fredenhagen and Rejzner, as well as Rejzner's thesis: <a href="http://arxiv.org/abs/1101.5112" rel="nofollow">arXiv:1101.5112</a>, <a href="http://arxiv.org/abs/1110.5232" rel="nofollow">arXiv:1110.5232</a>, <a href="http://arxiv.org/abs/1111.5130" rel="nofollow">arXiv:1111.5130</a>.</p>