VLA proper motion constraints on the origin, age, and potential magnetar future of PSR J1734$-$3333

The characteristic age of PSR J1734$-$3333 estimated from its current spin down rate implies that it is a young pulsar ($\tau_c<10$ kyr). But the time derivative of its spin down rate differs markedly from that assumed for normal radio pulsars, meaning its actual age is uncertain. G354.8$-$0.8 is a supernova remnant (SNR) whose centre is located 21 arcmin away of the pulsar, and with a morphology that suggests an association with the pulsar. We want to assess the likelihood of the association between PSR J1734$-$3333 and G354.8$-$0.8 or other nearby supernova remnants quantitatively, with the objective of shedding light on the real age of this pulsar. Observations with the Karl G. Jansky Very Large Array were carried out in 2015 and 2019 that allow precise astrometric measurements and consequently a proper motion estimate for the pulsar. The proper motion was found to be $\mu_\alpha=10\pm10$ mas yr$^{-1}$ and $\mu_\delta=-29\pm11$ mas yr$^{-1}$ (error bars are $1$-$\sigma$). Though marginal, this detection rules out the association with G354.8$-$0.8 because it means the pulsar is not moving away from the centre of the SNR. No SNR consistent with the measured proper motion and an age $\sim\tau_c$ could be found. We also present the first measurement of the spectral index for this pulsar, $\alpha=-1.1\pm0.3$, measured between $1.5$ and $3.0$ GHz. The SNR produced by the birth supernova of PSR J1734$-$3333 could have already faded to undetectable brightness, for which estimates suggest timescales of $10$-$100$ kyr. This and other considerations lead us to conclude that the pulsar is possibly older than $45$-$100$ kyr. PSR J1734$-$3333 is a pulsar with rotational properties that place it between standard radio pulsars and magnetars, and we interpret our result in the context of a possible future life as a magnetar for this pulsar.