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Visit Stack ExchangeAccording to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2,$$$$U_tU_s=U_{t+s}\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$$$U_tU_s=\omega(t,s)U_{t+s}\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_tU_s=U_{t+s}\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only$$U_tU_s=\omega(t,s)U_{t+s}\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1$$$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$, thethe transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2$$$$U_{t+s}=U_tU_s\tag 2,$$, thethe mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1$$, the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2$$, the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
According to the Schrödinger equation$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$the transformation$U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every$t\in\mathbb R$ is a unitary transformation of the Hilbert space$\mathcal H$, and since from this equation,
$$U_{t+s}=U_tU_s\tag 2,$$the mapping,$t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of$\mathcal H$, but they are elements of the projective Hilbert space$\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective group, that is, by Wigner's theorem, instead of (2), only$$U_{t+s}=\omega(t,s)U_tU_s\tag 3$$must hold, where$\omega(t,s)$ is a complex number of modulus$1$ depending on$t$ and$s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
