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Hi! In section 3.2 of Ballentine, the author considers the most general effect of a Galilean transformation on space-time coordinates ##\{\mathbf{x}, t \}## (3.5) and denotes such a transformation by ##\tau = \tau (R,\mathbf{a},\mathbf{v}, s)## where ##R## is a rotation matrix, ##\mathbf{a}## is a space displacement, ##\mathbf{v}## is a 3-velocity, and ##s## is a time displacement. Based on his later notation directly below, ##\tau \{\mathbf{x}, t \} = \{\mathbf{x'}, t' \}## it seems like ##\tau## is a map ##\tau : \mathbb{R}^{3}\times \mathbb{R} \rightarrow \mathbb{R}^{3}\times \mathbb{R}##.
He then says that ##\tau## must have corresponding to it a unitary transformation ##U(\tau)## with ##|\psi \rangle \rightarrow U|\psi \rangle ## and ##A\rightarrow UAU^{-1}## as usual (top of p.67). Furthermore, the Galilei group has ten parameters, ##s_{\mu}## (the 3 rotation angles, 3 velocity components, 3 spatial translations, and 1 temporal translation), and associated with each ##s_{\mu}## is a Hermitian generator ##K_{\mu}## such that ##U(s_{\mu}) = e^{i s_{\mu} K_{\mu}}##. At the inception of section 3.3 Ballentine notes that a unitary transformation ##U(\tau)## corresponding to a space-time transformation ##\tau## will have the most general form ##U(\tau) = \prod_{\mu=1}^{10}U(s_{\mu}) = \prod_{\mu=1}^{10}e^{is_{\mu}K_{\mu}}## (3.9).
What's confusing me is the functional form of ##U(\tau)## when compared to the functional forms of the ##U(s_{\mu})##. Each ##U(s_{\mu})## corresponds to a one-parameter family of unitary operators but ##\tau## seems to be an endomorphism of Galilean space-time so what does ##U(\tau)## correspond to? At first I thought it corresponded to a ten-parameter family of unitary operators ##U(s_1,...,s_{10})## since ##\tau = \tau (R,\mathbf{a},\mathbf{v}, s) = \tau(s_1,...,s_{10})## but the fact that he subsequently writes ##\tau \{\mathbf{x}, t \} = \{\mathbf{x'}, t' \}## has me confused as to what ##\tau## is as a mathematical object, what explicit functional dependence it has on the ##s_{\mu}##, and if I can actually call ##U(\tau)## a ten-parameter family of unitary operators.
Thanks in advance for clearing this up!
He then says that ##\tau## must have corresponding to it a unitary transformation ##U(\tau)## with ##|\psi \rangle \rightarrow U|\psi \rangle ## and ##A\rightarrow UAU^{-1}## as usual (top of p.67). Furthermore, the Galilei group has ten parameters, ##s_{\mu}## (the 3 rotation angles, 3 velocity components, 3 spatial translations, and 1 temporal translation), and associated with each ##s_{\mu}## is a Hermitian generator ##K_{\mu}## such that ##U(s_{\mu}) = e^{i s_{\mu} K_{\mu}}##. At the inception of section 3.3 Ballentine notes that a unitary transformation ##U(\tau)## corresponding to a space-time transformation ##\tau## will have the most general form ##U(\tau) = \prod_{\mu=1}^{10}U(s_{\mu}) = \prod_{\mu=1}^{10}e^{is_{\mu}K_{\mu}}## (3.9).
What's confusing me is the functional form of ##U(\tau)## when compared to the functional forms of the ##U(s_{\mu})##. Each ##U(s_{\mu})## corresponds to a one-parameter family of unitary operators but ##\tau## seems to be an endomorphism of Galilean space-time so what does ##U(\tau)## correspond to? At first I thought it corresponded to a ten-parameter family of unitary operators ##U(s_1,...,s_{10})## since ##\tau = \tau (R,\mathbf{a},\mathbf{v}, s) = \tau(s_1,...,s_{10})## but the fact that he subsequently writes ##\tau \{\mathbf{x}, t \} = \{\mathbf{x'}, t' \}## has me confused as to what ##\tau## is as a mathematical object, what explicit functional dependence it has on the ##s_{\mu}##, and if I can actually call ##U(\tau)## a ten-parameter family of unitary operators.
Thanks in advance for clearing this up!