# Can we make the heat equation covariant ?

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Good morning!

I was wondering if one can extend the heat equation in order to make it covariant.

In other words, in there some kind of operator $D_\mu$and a vector field (or perhaps another representation of the Lorentz group) $A^\mu$such that one can derive the heat equation $(\partial_0 - \partial^i \partial_i)\phi = 0$ as a projection or a special case of $D_\mu A^\mu = 0$

All I can find has a special treatment for time that I want to avoid.

Here are some random thoughts that did not actually help me :

• Perhaps I should add a length-dimensional factor $\alpha$ to $\partial^i \partial_i$ which has a certain transformation behaviour ;
• There is a problem with the number of Lorentz indices in the expression of the heat equation ;
• I somehow expect the heat/energy to diffuse with a wave-like front replacing the instant transmission of heat of classical solutions ;
• Heat is energy which is part of the 4-momentum : we should be able to find another 4-vector with heat as its timelike component ;
• Using the substitution $t\mapsto i\tau$, one can transform the heat equation into a Schrödinger equation ; the covariant heat equation thus might be found using a covariant extension of the Schrödinger equation, such as the Dirac equation (but I do not expect the Dirac equation to be the right one : what would spinors do here ?!)

Using the substitution tiτ, one can transform the heat equation into a Schrödinger equation ; the covariant heat equation thus might be found using a covariant extension of the Schrödinger equation, such as the Dirac equation (but I do not expect the Dirac equation to be the right one : what would spinors do here ?!)

Such a way of "obtaining" ("deriving") a Schrödinger equation is wrong. You should not rely on appearance because time stays time - it does not become "imaginary", but the equation variable $T(x,\tau)$ changes its meaning completely. And vice versa: such variable changes are not a right way of obtaining correct "heat equations".

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