HTTP just doesn't work that way. Aside from that, even if you could serialize a network connection (which you can't) it won't help you do what you want to do.
in reply to How to keep network connection persistent during session?
If you want to use HTTP, the best easily achievable goal is to use Keep-alive on the connection (which AFAIK any decent webserver should support / suggest / do already by default, but it may help to test it and/or increase the timeout. This won't help you with the authentication part, since HTTP doesn't guarantee that single connections serve only a single session - it's even possible that in between proxies could "bundle" multiple sessions over a single connection (though I don't know of any proxy that does that).
In any case, IME if authentication/reconnection takes such a large part of your system, you're either doing it wrong, or you're not doing anything very interesting.
If you're really that pressed for processing power, switch to a protocol that's a little less designed for single request/response actions and more designed for long-term connections. Like STOMP. Of course, that means ditching a pure web based approach...