I'm looking for a short story I read in an anthology in the 70s and I remember it was a pretty old book at the time.
"Schrödinger's Cat", a novelette by Rudy Rucker, matches your description. However, barring time travel, you could not have read it in the 70s, in an anthology which was "a pretty old book at the time", because it was first published in Analog Science Fiction/Science Fact, March 30, 1981. Could you possibly have read it in Rucker's 1983 collection The 57th Franz Kafka? Does any of these covers look familiar?
P.S. Thanks to the OP for pointing out that the story is available at the author's website.
In it someone invents a tube of sorts inside which time runs in reverse. He sets it on the floor and aims a toy train at it to run into the tube.
It's a Lego car, not a train:
The twins had brought the little car, a bright red-yellow-blue mass of Lego blocks. On the top was a battery-run motor, with a cogwheel linked by a black plastic chain to a gear on the single front wheel.
Klara examined our "time-tunnel" with interest. The core of it was the shoe-box-sized vacuum chamber made of phase-mirrors. You could see in quite easily. The thick loops of the guiding-field wires arched over the box like croquet wickets.
I removed the rifle from its mount on one end of the lab-table, and waited while Ion got the car from the little girls.
Then, bustling a bit, he lined up his three women in chairs against the wall, and set the car down at one end of the table. I cleared my throat, preparatory to telling them what they might expect, but Ion shushed me.
"First let them see, and then we'll discuss it."
I taped an iron nail to the bottom of the Lego car, and dialed the guiding-field's power up to some hundred times the level we had used before. The Lego car made a pretty big test-particle.
When the train is the same distance from the tube as the tube is long, a duplicate train appears at the far end, and they approach each other, face to face, till they pass through each other at the tube entrance.
In all frankness, I expected the experiment to be a failure. The car would roll up to the phase-mirror box, bump into the side and stop . . . nothing more. But I was wrong.
As the little car labored across the table towards the left end of the box, something happened at the right end. Seemingly out of no place, an identical Lego car pushed out of the right end of the tunnel and went chuffing on its way! "And there's one inside now, rolling left!" Klara exclaimed, leaning forward. She was right. For a few seconds there were three Lego cars on the table.
Car (1): The original car, still approaching the tunnel's left entrance. Car (2): The one moving in the tunnel, from right to left. Car (3): The new one moving away from the right end of the tunnel.
And then car (1) and car (2) met at the left-end mirror. They melted into each other . . . nose into nose, wheel into wheel, tail into tail. It was like watching a Rorschach ink-blot disappear into its central fold.
One of the twins squealed and ran to catch car (3) before it ran off the other end of the lab table. I took it from her and examined it closely. Car (3) appeared to be identical to car (1). We had already done this experiment with electrons and with small bullets . . . but one bullet or electron is much like another. Until now I had been unwilling to accept Ion's interpretation of our experiment. But it certainly looked as if car (3) really was car (1).
Then someone does it again, but this time, when the second train appears in the tube, and the two are heading for each other, he grabs the outside train to prevent it from entering the tube, creating a paradox, to see what happens.
Ion had conducted a third experiment. The car was to roll towards the tunnel while he watched both ends. His plan was to stop car (1) if car (3) appeared, and to let car (1) go if car (3) did not appear. This meant that a car would come out of the right end of the tunnel if and only if no car came out of the right end of the tunnel. Yes if and only if no.
The immediate result is his head splits in two,
Question: When Ion actually ran the experiment, did car (3) appear? Answer: Yes and no.
I closed the lab book and looked around the room. The scattered bits of Legos . . . how many?
"What happened, Ion? Did the car come out of the tunnel?"
"Yes," Ion said, raising his head from on top of his arms.
"No," Ion said, uncrossing his arms and raising up his other head from under the arms.
The two faces looked at me, each of them a bit translucent, a bit unreal. The two necks merged into his collar, making a solid, tubular letter "Y."
[. . . .]
"I'm in a mixed state, William. I ran the paradox. It had to come out both ways." He turned the switch to power-up the guiding-field. It was dangerous to be restarting it without a vacuum in the chamber.
then four, repeating until his head is a myriad tiny stalks, killing him.
There was a crash behind me. I whirled around. The time-tunnel was billowing smoke and the phase-mirrors had smashed into pieces. For a second I couldn't see Ion through the smoke, but then he came at me.
A tangle of twenty or a hundred thin necks writhed out of his open collar, and on the end of each tentacle-like neck rode a tiny grimacing head, and every little head was screaming at me in a terrible tiny voice. . . .
He dispersed completely after that. As different variants of Ion Stepanek split off into different universes, each corresponding head would shrink . . . get "farther away" . . and a copy of his body would split off with it, twisting and dwindling. I don't know how long it took; and I don't know how I could have seen it; I wish I could forget it. The horrible squid-bunch of necks, each little head screaming out something different . . . I hope he's really gone.
