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Quantum Physics
Time/space symmetry in Dirac Equation
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[QUOTE="Physics Footnotes, post: 5505604, member: 597438"] This is a terrific question, and one which puzzled me for a long time too! Here's how I stopped losing sleep over it... Let's start out with the Lorentz-covariant form of the Dirac Equation as you have given it: $$i\gamma^\mu\partial_\mu\Psi(\mathbf {x},t)=mc\Psi(\mathbf {x},t)$$ For clever choices of ##4 \times 4## matrices ##\mathbf{\alpha}=(\alpha_1,\alpha_2,\alpha_3)## and ##\beta## we can rewrite this in the form: $$i\frac {\partial}{\partial t}\Psi(\mathbf {x},t)=(-ic\mathbf {\alpha} \cdot \nabla + \beta mc^2)\Psi(\mathbf {x},t)$$ By defining $$H_0:=-ic\mathbf {\alpha} \cdot \nabla + \beta mc^2$$ we can write this equation in the form $$i\frac {\partial}{\partial t}\Psi(\mathbf {x},t)=H_0\Psi(\mathbf {x},t)$$Whilst this is getting close to the Hilbert Space formulation, we mustn't jump to conclusions because our solutions ##\Psi(\mathbf{x},t)## are functions of space [I]and [/I]time instead of functions of space [I]parametrized by[/I] time. To get Schrodinger mechanics, we carefully separate out space and time with a scalpel ##\psi## (lower case psi this time!) defined as follows:$$\psi(t):=\Psi(\cdot,t)$$ It turns out that for each time ##t##, ##\psi(t)## is not only a function of ##x##, but is square-integrable with respect to the following Hilbert Space (depending on which way you like your eggs): $$\mathscr{H} := L^2(\mathbb{R})^4\equiv L^2(\mathbb{R}^3,\mathbb{C}^4) \equiv L^2(\mathbb{R}^3) \otimes \mathbb{C}^4$$ Furthermore, since our operator ##H_0## defined above does not depend either explicitly or implicitly on time, it can safely be interpreted as an operator in ##\mathscr{H}##. (I'll leave out some subtleties about the precise domain on which ##H_0## must be defined as this doesn't impact on your question.) Finally, we can now write $$i\frac {d}{dt}\psi(t)=H_0\psi(t)$$ where, for any instant of time ##t##, ##\psi(t)## is a function on ##\mathbb{R}^3## (into ##\mathbb{C}^4##), and is interpreted as the (spatial) wave function of the electron at time ##t##. The reason I went through all that is to convince you that it is indeed true that a Lorentz-covariant differential equation can be re-expressed in a form that looks completely asymmetric in space and time. That's why we often use the modifier 'manifestly' to describe equations that are purposely written in a space/time symmetric way. It is (to me at least) extremely surprising to first find out that the Hilbert Space formalism (which feels so non-relativistic) can accommodate both a relativistic and a non-relativistic treatment of a system, but it turns out to have a compelling explanation. Hilbert Spaces are flexible enough to carry representations of both the Galilean group and the Lorentz-Poincare group (and pretty much any other algebraic invention actually!). Classical state spaces don't have that linear structure. Another good way to reconcile this state of affairs is to revisit good old Maxwell's Equations. We first learn these equations in a very non-relativistic-looking way by considering E-B fields defined on space but which vary in time. Yet, they can be re-expressed in the form of a simple (coordinate-free) tensor field on Minkowski spacetime. You can't get more manifestly Lorentz-covariant than that; there aren't even any transformations to check! [/QUOTE]
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Time/space symmetry in Dirac Equation
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